I
NASA TECHNICAL
TRANS L ATION
On
»•»
HELICOPTERS
CALCULATION AND DESIGN
Volume I. Aerodynamics
by M. L. MiP et al.
"Mashinostroyeniye" Publishing House
Moscow, 1966
NASA
TT
v.l
c.l
11 NASA TT F-494
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • SEPTEMBER 1967
TECH LIBRARY KAFB, NM
DDba^71
NASA TT F-494
HELICOPTERS
CALCULATION AND DESIGN
VoL I. Aerodynamics
By M. L. Mil', A. V. Nekrasov, A. S. Braverman,
L. N. Grodko, and M. A. Leykand
Translation of "Vertolety. Raschet i proyektirovaniye . 1. Aerodinamika."
Izdatel'stvo Mashinostroyeniye, Moscow, 1966.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
For sale by the Clearinghouse for Federal Scientific and Technical Information
Springfield, Virginia 22151 - CFSTI price $3.00
MNOTATION /g'
The work "Helicopters (Calculation and Design)" is published in three
volumes .
Vol.1 - Aerodynamics;
Vol.11 - Vibrations and Dynamic Strength;
Vol. Ill - Design.
The first volume is devoted to ways of developing helicopters, the basic
principles of their design, and the position occipied by helicopters among other
means of aviation not requiring airfields. Various theories of rotors and cor-
responding methods of determining their aerodynamic characteristics are pre-
sented: the classical theory of a rotor with hinged blades in the general case
of curvilinear flight of the helicopter; the momentum theory of an ideal rotor
and its application to the energy method of calculation; the classical theory
when using methods of numerical quadrature; the vortex theory and methods of
experimental determination of rotor performance in flight tests and in wind
tunnels. Various methods of aerodynamic calculation of a helicopter and the
theory of blade flutter are presented in detail. This volume gives an account
of methods of calculating flutter in hovering and in forward flight. Particular
attention is devoted to consideration of friction in the axial hinges of the hub
and to the transfer of blade vibrations through the automatic pitch control
mechanism. Experimental investigations of flutter are described.
The book is intended for engineers of design offices, scientific workers,
graduate students, and teachers of higher institutes of learning. It might be
useful to engineers of helicopter manufacturers and to students for furthering
their knowledge of the aerodynamics and mechanical strength of helicopters.
Mary sections of the book will be a useful tool also to flight and technical
staffs of helicopter flight units.
■«•
^ Numbers in the margin indicate pagination in the original foreign text.
11
PREFACE 1^
The present book generalizes the experience of the scientific work and
practical design activity of engineers of one of the Soviet teams working on the
development of helicopters.
Twenty years ago, when the team had just set out on their work, everything
in this field seemed to have been already long discovered and in-vented.
Those to whom belongs credit for the original ideas and designs of rotary-
wing aircraft - Leonardo da Vinci, M.V.Lomonosov, M .Ye . Zhtikovskiy (Joukowski),
B.N.Yur'yev, and others - had long ago proposed almost all of the existing de-
signs of helicopters. Designers, scientists, and inventors in -various countries
birilt dozens of helicopter models which successfully rose into the air. However,
not one of these rotocraft was suitable for practical use, large-scale produc-
tion, or reg\£Lar ser-vice.
A very difficult problem that required considerable and tedious work re-
mained ixnsol-ved, namely, the problem of de-veloping helicopters which would find
practical use in everyday life.
To solve this problem we had at our disposal an inportant scientific basis
in the form of classical works, the studies of the Central Aero-Ifydrodynamic
Institute (TsAGi), and of foreign scientists. However, testing of each new air-
craft confronted design engineers with new acute problems and forced them to
work out many theoretical problems to find the proper method of sol-ving specific
design problems.
This volume discusses the basic problems of the theory, calculation, and
design of helicopters worked out by the team and representing the vital interests
of its design acti-vity.
The fact that some of the authors had occasion to participate in applying
the classical rotor theory to the calculation and design of the first autogiros,
in the original experimental work on models and on full-scale rotors in wind
tunnels, in de-veloping methods of aerodynamic calculation of helicopters, and
then - for more than 15 years - in designing an entire family of helicopters of
the same configuration in all weight classes, offers an opportunity to elucidate
the basic problems of the theory and calculation of helicopters that have been /4
checked out by practice.
As early as 194S there was not a single helicopter in service in our
country. Now thousands of such machines created by various design teams assist
people in many areas of their life and activity.
Engineers and designers working on the design or construction of heli-
copters, pilots and technicians, students of air academies who are studying or
are interested in helicopters will find useful information in this book.
iii
Engineering, especially aircraft engineering, is rapidly becoming obsolete.
However, it is hoped that the general methods of approach to the development of
a new type of aircraft, as presented in this book, will outlive today's heli-
copter models.
M.Mil'
Chapter I of Vol.1, Sections 1 and 2 of Chapter II, and Section 2 of Chap-
ter III were written by M.L.Mil'; Chapter I? and Section 5 of Chapter II were
written by A.V.Nekrasov; the remaining Sections of Chapters II and III and also
Subsections 19-28 of Section 2 of Chapter II were written by A.S.Braverman.
In preparing the manuscript, the authors were assisted by engineers F.L.
Zarzhevskaya, R.L.Kreyer, and L.G.Rudnitskiy.
Reviewer R.A.Mikheyev made many valuable comments.
The authors express their sincere gratitude to these coworkers.
XV
TABLE OF CONTENTS
Page
Preface ill
Notations xLii
CHAPTER I EVOLUTION HISTORY OF HELICOPTERS AND BASIC
DESIGN PRINCIPLES 1
Section 1. Evolution of the Helicopter Industry 1
1. Development of Helicopters in Size 3
2. Qualitative Development of Helicopters 8
3. Special-Purpose Helicopters 13
4. Compound Helicopters with Additional
Engines - Rotocraft 15
Section 2. The Helicopter Compared to Vertical Takeoff
and Landing and Short Takeoff and Landing
Aircraft l6
1. Tactical and Technical Requirements for VTOL
and STOL Military Transport Aircraft of the West .. 17
2. Means for Increasing the Flying Range of
Helicopters 21
3. Helicopter with Takeoff Run 23
4. Takeoff Distance of Helicopter 25
5. Criterion for Estimating the Economy of
Various Transport Aircraft 27
6. Possibilities of Increase in Maximum Fljring
Speed 31
Section 3« Basic Principles of Design 33
1. Selection of Engine Horsepower and Rotor Span ... 33
2. Analysis of Multirotor Configurations 39
CHAPTER II ROTOR AERODYNAMICS 45
Section 1. Development of Rotor Theory and Methods of
Experimental Determination of its Characteristics ... 45
1. Classification of Rotor Theories 54
2. Development of Experimental Methods 54
Section 2. Classical Theory of a Rotor with Hinged Blade
Attachment; General Case; Curvilinear Motion 56
Rotor Theory in Curvilinear Motion 57
1. Coordinate System and Physical Scheme
of the Phenomenon • 57
2. Inertia Forces Acting on the Blade 59
3. Aerodynamic Forces Acting on the Blade 65
4. Equation of Moments Relative to Flapping
Hinge .« 66
5. Physical Meaning of the Obtained Result 70
6 . Equation of Torque 72
■■■■■■ ■■■■■■
Page
7. Rotor Thrust and Angle of Attack 74
8. Lateral Force 75
9 . Longitudinal Force 77
10. Consideration of the Change in the Law of
Induced Velocity Distribution during
Curvilinear Motion 7S
Analysis of Obtained Results 83
11. Blade Flapping 83
12. Effect of Curvilinear Motion at Autorotation
of the Rotor 86
13. Behavior of the Resultant of Aerodynamic
Forces in Curvilinear Helicopter Motion 88
Effect of Rotor Parameters and Hub Design on Flapping
and Damping of the Rotor 91
14. Rotor with a Profile Having a Variable
Center of Pressure 91
15. Effect of Blade Centering 92
16. Rotor with Flapping Compensator 94
Rotor Flapping in Curvilinear Motion of the Rotor Axis
at Variable Angular Velocity 96
17. Uniformly Accelerated Rotation of the
Rotor Axis 96
18. Harmonic Oscillation of the Rotor Axis 100
Characteristics of Rotor Aerodynamics Determined by
Hinged Blade Attachment 102
19 . Physical Meaning of Blade Flapping 103
20. Redistribution of Aerodynamic Forces over
the Rotor Disk due to Flapping 104
21. Approximate Derivation of Formulas for
Flapping Coefficients 107
22. Effect of Nonuniformity of the Induced
Velocity Field on the Flapping Motion 109
Method of Calculating the Aerodynamic Characteristics of
a Rotor for Azimuthal Variation of Blade Pitch 114
23. Equivalent Rotor Theory 114
24. Derivation of Formulas for a Rotor with
Flapping Hinges as for a Rotor without Hinges.
Conditions of Equivalence of Hinged and
Rigid Rotors 123
25. General Expressions for Determining the Com-
ponents of Blade Pitch Change cfb , cpi , and cpi 132
26. Determination of Flapping Coefficients of
Rotor with Flapping Compensator 138
27. Determination of the_ Components of Blade
Pitch Change cpi and 91 after Deflection of
the Automatic Pitch Control 140
28. Sequence of Aerodynamic Calculation of a
Rotor with Variable Pitch •...*. 144
Section 3 . Momentum Theory of Rotor 14^
1. Theory of an Ideal Helicopter Rotor 147
VI
Page
2. Derivation of the Expression for the Torque
Coefficient of a Real Rotor 156
3. Rotor Profile Losses ...« 160
4. Certain Considerations in Selecting Blade
Shape and Profile l64
5. Approximate Determination of Rotor Profile
Losses *. 169
6. Effect of Air Compressibility of Rotor
Profile Losses 170
7. Induced Losses of a Real Rotor 178
8. Determination of Angle of Attack and
Pitch of Rotor 183
Section 4- Classical Rotor Theory. Method of Numerical
Integration • 184
1. Formulas for Calculating Forces and Moments
of a Rotor 185
2. Method of Calculation 193
3. Aerodynamic Characteristics of Profiles
for Rotor Blades 195
4. Distribution of Aerodynamic Forces over
the Rotor Disk 200
5 . Aerodynamic Characteristics of Rotor 206
6. Aerodynamic Characteristics of Rotor in
Autorotation Regime 209
7. Limit of Permissible Helicopter Flight
Regimes (Flow Separation Limit) 212
8. Distribution of Profile Losses over Rotor Disk.
Dependence of Profile Losses on Aerodynamic
Characteristics of Blade Profiles 218
Section 5. Vortex Theory of Rotor 222
1. Problems in Vortex Theory 222
2. Theoretical Schemes for the Vortex Theory of a
Rotor with a Finite Number of Blades 224
3. Form of Free Vortices 226
4. Determination of the Induced Velocities by
the Biot-Savart Formula 227
5. Use of the Biot-Savart Formula in Developing
the Vortex Theory of a Rotor 228
6. Axial Component of Induced Velocity from
Bound Vortices 230
7. Axial Component of Induced Velocity from
Spiral (Longitudinal) Vortices 230
8. AicLal Component of Induced Velocity from
Radial (Transverse) Vortices 232
9. Integrodifferential Equation of the Vortex
Rotor Theory 232
10. Constancy of Circulation of Trailing Vortices
along Straight Lines Parallel to the Axis of
the Inclined Vortex Cylinder and Possible
Simplifications 234
vxx
Page
11. Characteristics of Using the Lifting-Line
Scheme and Scheme of a Vortex Lifting Siirf ace .... 236
12. Division of Vortices into Types Close to
and Remote from the Blade; Use of "Steady-
Flow Hypothesis" 237
13. Instantaneous and Mean Induced Velocities
and Generation of Variable Aerodynamic
Loads on the Blade 238
14. Characteristics of the Extrinsic Induced
Velocity Field 238
15. Vortex Theory of a Rotor mth an Infinite
Number of Blades 239
Vortex Theory of Wang Shi-Tsun 240
16 . Rotor Scheme 240
17. Determination of Induced Velocities 241
18. Calculation Formulas for Induced Velocity
Determination 241
19. Application and Evaluation of the Possibilities
of the Wang Shi-Tsun Vortex Theory 243
Vortex Theory of V.E.Baskin 244
20. Scheme of Rotor Flow 245
21. Determination of Induced Velocities from
the Dipole Column 246
22. Fluid Flow Induced by a Disk Covered with
Dipoles 247
23 . Boundary Conditions 249
24. Transformation of Eq.(5.67) to the Rotor Axes;
Use of the Theorem of Addition of Cylindrical
Functions 249
25. Determination of the Total Velocity Potential
from the Entire Dipole Column 250
26. Determination of Induced Velocities 252
Section 6. Experimental Determination of Aerodynamic
Characteristics of a Rotor 253
1. Flight Tests for Determining the Aerodynamic
Characteristics of a Helicopter 254
2. Wind- Tunnel Tests for Determining the
Aerodynamic Characteristics of a Rotor 257
Methods of Converting the Aerodynamic Characteristics
of a Rotor 26l
3. Conversion of Aerodynamic Characteristics to
a Different Rotor Solidity Ratio 26l
4. Conversion of Aerodynamic Characteristics on
Variation in Minimum Profile Drag Coefficient
of the Blade Sections Cxpg • 265
5. Conversion of Aerodynamic Characteristics on
Variation in the Peripheral Speed of the
Rotor (Mo Numbers) 266
6. Conversion of Angle of Attack and Rotor Pitch
vxxi
Page
on Variation in Inclination of the Automatic
Pitch Control, Flapping Compensator, and Mass
Characteristic of the Blade 26?
7. Examples of Using the Conversion Formulas 268
Section ?• Performance and Propulsive Efficiency Coeffi-
cient of a Rotor 270
1. Performance and Efficiency of Rotor
Proposed by K .Khokhenemzer 271
2. Determination of Performance and Propulsive
Efficiency of a Rotor 273
3. Performance and Efficiency of a Rotor,
Obtained from Experimental Data ........*.. 277
4. Performance and Efficiency of a Rotor,
Obtained from Calculated Graphs 279
5. Conversion of Performance and Efficiency on
Variations in Rotor Parameters 282
6. General Comments on Rotor Efficiency and
Performance 283
Section 8. Calculation of Rotor Characteristics in
Hovering and Vertical Ascent (Momentum Theory
of Propellers) 284
1. Brief Review of the Momentum Theory of
Propellers 284
2. Results of Calculating the Characteristics
of a Rotor 286
3. Approximate Method of Determining the
Dependence of mt on t 292
4. Conversion of Aerodynamic Characteristics on
Variation in the Rotor Solidity Ratio 295
5. Determination of Optimal Aerodynamic Parameters
of a Rotor with Consideration of the Dependence
of Characteristics on Mo 296
CHAPTER III AERODYNAMIC DESIGN OF A HELICOPTER 301
Section 1. Basic Equations for Aerodynamic Design of
a Helicopter 301
1. Aerodynamic Design Principle of a Helicopter .... 301
2. Equation of Motion of a Helicopter 301
3. Various Methods of Determining Aerodynamic
Rotor Characteristics and Methods of
Aerodynamic Design 303
4. Calculation of Composite and Multirotor
Craft 304
5. Induction Coefficients of Two-Rotor Helicopters
and Helicopters with a Wing 308
Section 2. Aerodynamic Helicopter Design by the
Mil'-Yaroshenko Method 315
1. Equations of Motion and Design Principles 315
2. Determination of Aerodynamic Rotor Characteristics . 318
IX
Page
3. Calculation of Flight Data 320
4. Limits of Applicability of the Method 322
Section 3. General Method of Aerodynamic Design for
Rotary Wing Aircraft 323
1. Construction of A-uxiliary Graphs for Helicopter
Performance Data 324
2. Determination of Helicopter Performance Data .... 331
3. Graphs for Determining Optimum Helicopter
Aerodynamic Parameters 342
Section 4. Aerodynamic Design of a Helicopter Based on
Concepts of Rotor Performance and Efficiency 346
1. Helicopter Performance 347
2. Performance of Multirotor and Composite
Helicopters 348
3. Determination of Helicopter Flight Data 357
4. Calculation of a Helicopter -with a Tractor
Propeller 363
5. Comparison of Helicopter and Airplane 364
6. Power of Front and Tail Rotors in a Helicopter
of Fore-and-Aft Configuration 366
7. Retraction of Landing Gear on Helicopters 368
Section 5. Aerodynamic Calculation of a Helicopter by
the Power Method 369
1. Determination of Required Power in Horizontal
Helicopter Flight 370
2. Determination of Helicopter Performance Data .... 375
3. Relation between Np^, Ni„4, and Npaj. during
Horizontal Flight of a Single-Rotor Helicopter , . . . 376
CHAPTER IV ROTOR FLUTTER '. . . . 379
Section 1. Basic Assumptions and Characteristics of an
Approach to Flutter Calculation 380
1. Bending and Torsional Vibrations of the Blade.
Possible Cases of Stability Loss 380
2. Effect of Blade Attachment to Hub and the
Possibility of Theoretical Investigation of
Flutter of an Isolated Blade 381
3. Different Types of Flutter Differing with
Respect to Blade Vibration. Flapping and
Bending Flutter 381
4. Characteristics of the Torsional Vibration
Modes of a Blade and Possible Correlated
Assumptions 382
5. Assumptions on Blade Oscillations in the Plane
of Rotation 383
6. Determination of Aerodynamic Forces Acting on
a Vibrating Profile 384
Section 2. Flapping Flutter of an Isolated Blade with
Axial Flow past the Rotor 386
Page
1. Blade Model 386
2. Derivation of Differential Equations
of Flutter 3S7
3. Particular Solution of the Differential
Equation 391
4. Differential Equation of Disturbed Motion 391
5. Notation of Differential Equations in
Matrix Form 392
6. Solution of Differential Equations of
Blade Vibrations 392
7. Determination of the Critical Flutter Rpm 395
8 . Blade Divergence 396
9. Parameters Characterizing Blade Balance
(Effective Blade Balance) 396
10. Dependence of Critical Flutter Rpm on Blade
Balancing and Values of the Flapping Com-
pensator Coefficient 398
11. Blade Arrangement 399
12. Effect of Control Rigidity 400
13 . Conditions for Ab sence of Flutter 400
14. Mechanism of Generation of Forces Exciting
Flutter 401
Section 3- Consideration of Friction Forces during Flutter .... 406
1. Character of the Effect of Friction Forces
during Flutter 406
2. Linearization of Friction Forces 40?
3. Determination of Flutter Speed with
Consideration of Friction 408
4. Effect of Forced Motion in the Feathering
Hinge 409
Section 4- Rotor Flutter with Consideration of Coupling
of Blade Vibrations through the Automatic
Pitch Control 414
1. Forms of Rotor Flutter Observed in
Helicopter Experiments 414
2. Analytical Expression for Cyclic Modes of
Rotor Vibration 414
3. Cyclic Vibration Modes in Specific Cases
and Control Loads 416
4. Differential Equations of Rotor Flutter with
Consideration of Coupling of Blade Vibrations
through the Automatic Pitch Control 418
5. Transformation of Eqs.(4.18) in Particular
Cases where Cyclic Modes are the Solution of
the Differential Equations of Rotor Flutter .... 421
6. Rotor Flutter in the Presence of Different
Rigidity of Longitudinal and Lateral Controls ... 422
Section 5- Flapping Flutter of a Rotor in Forward Flight 424
1. Preliminary Statements 424
2. Differential Equations of Blade Oscillations
in Forward Flight 424
XI
3. Solution of Differential Equations
4. Determination of Critical Flutter Rpm
"without Consideration of Harmonic Components
of Blade Motion
5. Effect of Flying Speed on Critical
Flutter Rpm
Section 6. Calculation of Flutter vd_th Consideration of
Bending and Torsion of the Blade
1. Bending and Torsion of Blade during Flutter
2. Determination of the Torque from Bending
Forces on the Blade
3. Differential Equations of Binary Vibration
4. Solution of Differential Equations
5. Calculation of Flutter with Consideration
of Three Degrees of Freedom
6. Calculation of Flutter with Three Degrees
of Freedom Disregarding Blade Torsion
7. Calculation Results
8 . Bending Flutter
9. Approximate Method of Determining the Mode
of Bending Vibrations in Flutter
Section ?. General Method of Calculation of Flutter and
Bending Stresses in the Rotor Blade during
Flight
1. Calculation Method and its Possibilities ...
2. Basic Assumptions and Suggestions
3 . Differential Equations
4. Boundary Conditions of the Problem
5. Determination of Equivalent Rigidity of the
Control System ,
6 . Determination of Aerodynamic Forces ,
7. Method of Solving the Differential Equations
8. Transformation of Partial Differential
Equations into Ordinary Differential
Equations ,
9. Determination of the Magnitude of the Moment
of Friction in the Feathering Hinge of the Hub
10. Sequence of Performdng the Calculation
Section 8. Experimental Investigations of Flutter
1. Ground Tests for Flutter
2. Flutter Tests in Flight
3. Comparison of Calculation and Experiment
under Conditions of Axial Flow past the Rotor
4. Comparison of Calculation and Experiment in
Flight
5 . Check for Flutter
6. Experimental Determination of Control System
Rigidity
7. Experiments on Djmamically Similar Models ...
References
Page
426
427
428
429
429
430
432
434
436
441
446
447
450
452
452
454
454
455
456
457
460
462
466
467
470
470
475
478
480
480
481
483
487
Xll
NOTATIONS l^
Ae rodynajnic Characteristics
a = angle of attack of rotor;
ffp = angle of attack of blade section;
Qfo = angle of zero lift of blade profile;
Aa = downwash angle of flow;
^^^ = Cy = tangent of angle of slope of the lift curve
■with respect to angle of attack of the profile;
da
= tan"'' — ^ = inflow angle in blade section;
r = circulation in blade section;
M = Mach number of blade section;
Mq = average Mach number with respect to azimuth,
in tip section of blade (Mq = -—■
Mfi = flight Mach number (Mfi= -|-
Q
T, t = — I- = thrust and coefficient of thrust of rotor
/ _ T ^
l^ ~ i PctttR2((juR)^ J '
Q
H, h = —^ = longitudinal force and coefficient of longi-
tudinal force of rotor ( h =
^ paTTR^((juR)^
Cs
S, s = = lateral force and coefficient of lateral force
a
of rotor ( s =
i parrR^((i}R)=
m
M^, mt = - = torque and torque coefficient of rotor
Mt
m. =
•J- parrR^CouR)^
N = power of motor (of rotor in Chapt.II);
lift and lift coeffic:
(t ^ V
V ^ i nrrTTR2C.,R;i2 J '
Cv
Y, ty = —-I— = lift and lift coefficient of rotor
parrR^CcuR)^
xiix
Velocities
X, tx = — ^ = propulsive force and coefficient of pro- ^
( X \
pulsive force of rotor it, = — — — — =);
Cy» Cxp = coefficients of lift and profile drag of
blade section (airfoil) referred to dynamic
pressvire ^ pV^;
B = coefficient of tip losses;
cbR = angular velocity;
V, fv = -^ = path velocity of helicopter flight;
\ (bR /
\ (bR
^x» ^y» ^z ~ horizontal, vertical, and lateral conponents
of flight velocity;
V, ( V = ~ ^ - ■ ) = induced velocity;
\ luR /
U, ,^U = ^^1 = relative velocity of flow past a blade
V U)R / element;
Ux, Uy, (Ux = — ^, Uy = — 5L_| = horizontal and vertical conponents of rela-
^ '"-^ "^^ ' tive velocity of flow past a blade element;
\ = coefficient of flow;
p, = characteristic (coefficient) of rotor per-
formance .
Geometric Characteristics
D = diameter of rotor;
R = radius of rotor;
F = disk area; ,_ .
r = radius of rotor blade section f r = -^)>
b = blade chord ', b = -— -, b = -,- \
Zi, = number of blades;
Zr = number of rotors;
a = ^^^o-"^ = load factor of rotor;
ttR
^.h* ^v.h - distance from axis of rotation of rotor to
the horizontal (flapping) and vertical
(drag) hinges, respectively;
c = thickness of profile section, c = -^;
P = flapping angle of blade;
xiv
a„, b„ = coefficients of flapping;
9, 9o = blade angle (pitch); angle between chord of blade
profile and plane of rotation;
H, T) = angles of deflection of automatic pitch control mecha-
nism; H -with index = mutual influence coefficient of
lifting elements;
00 = blade angle at r = 0.7 for P = k = Tl = 0;
3i> 9s = cQDponents of change of blade angle relative to the
plane of rotation, due to deflection of the automatic
pitch control mechanism;
V = change of blade angle due to elastic deformation of
blade.
XV
HELICOPTERS; CALCULATION AND DESIGN. VOL.1: AERODINAMICS /2
M.L.Mil', Editor
ABSTRACT. A review of the historical development of Russian
and Western helicopters, in size and lift capacity, for civil
and military purposes is followed by detailed discussions on
rotor aerodynamics for various angles of attack, blade setting,
flapping angle, center-of-pressiire position, blade vibration
(natioral, forced, harmonic, etc.), and other rotor parameters
in their influence on rotor rpm and craft stability. Formulas
are given for the forces and moments of rotor danping in hover-
ing and forward flight; for the redistribution of aerodynamic
forces over the rotor disk due to flapping; for cyclic pitch
change of rotors with variable and constant pitch. The theory
of an ideal helicopter is developed on the basis of optimum
blade profile, prevention of rotor profile losses, and proper
balancing. Flutter in hovering and forward flight is calcu-
lated, with eirphasis on friction in the axial hub hinges and
transfer of vibrations through the automatic pitch control.
CHAPTER I
EVOLUTIONAL HISTORY OF HELICOPTERS AND BASIC
DESIGN PRINCIPLES
(Selection of Parameters and Configuration)
Section 1. Evolution of the Helicopter Industry
Designing is always directed toward the futiire. However, for a better
picture of the potentialities of the future development of helicopters it is
useful to attenpt to understand the basic trends of their evolution from past
experience. Natiu-ally, we are not interested here in the prehistory of heli-
copter construction, which we will only briefly mention, but in its history from
the time when the helicopter as a new type of aircraft became usefiil for practi-
cal application.
The writings of Leonardo da Vinci going back to 1483 contained the first
mention of an apparatus with a vertical rotor, a helicopter. The first stage of
evolution ranges from the model of a helicopter developed by M.V.Lomonosov in
1754 through a long series of designs, models, and even full-scale apparatus
which were not destined to rise into the air, to the construction of the world's
first helicopter which, in 1907, was able to become airborne. This four-rotor
helicopter was constructed by the French designers Breguet and Riche. In 1923,
a passenger became airborne for the first time in the USA in a helicopter de-
signed by de Bothezat. The first world altitude record of a helicopter of 18 m
was set in 1930 on the Italian coaxial helicopter by d'Ascanio.
In Russia, a single-rotor helicopter was built in I9II, on the basis of
the scientific research by M.Ye.Zhukovskiy devxated to helicopter rotors, by a
groigj of his students headed by B.N.Yur'yev. The configurations of this machine
represent the basic scheme of the single-rotor helicopters used widely at
present. B.N.Yur^yev was able to resume this work only in 1925* In 1932, a
grotf) of engineers headed ty A.M.Cheremukhin constructed the helicopter TsAGI
1-EA (Fig. 1.1) which reached an altitude of 6OO m and stayed in the air for
18 min, which - for that time - was an outstanding achievement. It suffices to
say that the official altitude record established three years later on Breguet^s
new coaxial helicopter was only 180 m.
At this time there was a pause in the development of helicopters. A new
branch of rotary-wing aircraft came to the forefront, known as autogiros. The
idea of the autogiro, as an aircraft with a rotary wing (freely rotating air-
foil) never losing speed, occurred to the young Spanish engineer Juan de la
Cierva in the 1920s. At that time, conventional aircraft whose development had
been vigorous during the years of World War I and which, by then, carried /8
armament and thus had greater wing loading were troubled by a new problem of
spin, i.e., stalling. It appeared sinpler to develop a safe and sufficiently
perfected autogiro than to build a helicopter. The rotor, freely rotating due
to the relative flow, eliminated the need for conplex reduction gearing and
transmissions. The hinged attachment of the rotor blades to the hub used on
autogiros gave far greater strength to the blades and higher stability to the
autogiro. Finally, engine failure ceased to be a threat, as had been the case
in the first helicopters; the autogiro, with autorotating blades, had no diffi-
ciolty in landing at low speed.
Fig. 1.1 Helicopter TsAGI 1-EA.
Cierva, working in England, created several autogiro designs, the best
known of which was the C-30 autogiro which was produced as a pilot series.
Autogiros were also built in the USA by the Pitcairn and Kellett Companies and
in the Soviet Union at TsAGI by the designers I.P.Bratukhin, V.A.Kuznetsov,
N.I.Kamov, .N.K.Skrzhinskiy, M.L.Mil', and others.
The flying speed of Soviet autogiros in 1937 reached 260 km/hr. The A-7
autogiros designed by N.I.Kamov were used at the front during the first year of
World War II.
The great lift capacity of the rotating rotor gave the autogiro a short
ground run. Even though, a mechanical drive from the engine, for spinning the
rotor before takeoff, was used in this design to further shorten the takeoff
run. In the design of the British C-40 autogiro the rotor was given a spin-Tjp
before fUght to an rpm such that, at the instant of disengagement from the
engine - which, in forward flight, rotated the propeller - the machine, due to
the marked increase in pitch, took off without a run, rising vertically into the
air.
Only one step remained for the development of a true helicopter. And this
step, as is always the case in technology, was made almost simultaneously in
various countries. This was the beginning of the present development stage of
helicopters . It was started by flights • of the FW-61 helicopter designed by
Professor Focke in Germany (1937), the VS-300 helicopter designed by Sikorsky
in the USA (1939), and the "Omega" helicopter designed by I.P.Bratukhin in the
USSR (1940). All three of these helicopters used a hinged rotor capable of
autorotation, which had already become standard for autogiros .
World War II somewhat delayed the development of helicopters. They were
still unsuitable for practical use, and the ways and means for experimental
studies were limited. After the end of the war (1946 and 1947), large numbers
of designers and inventors invaded this new and promising area of development of
aviation engineering. Within a short time, literally dozens of new helicopter /2.
designs were created. This was a contest of the most diverse schemes and con-
figurations, generally of the single- or two-seater type and used mainly for
experimental purposes • Military agencies were the only users of this expensive
and conplex equipment. The first helicopters in various countries were used as
liaison and reconnaissance military aircraft.
In the development of helicopters, just as in many other areas of tech-
nology, one can clearly distinguish two trends of development: the quantitative
trend concerned with size of the machine and the almost simultaneous qualitative
trend concerned with inprovement of the craft within a certain size or weight
class . The former trend represents development with respect to lift capacity
and the second with respect to inprovement of the tactical or economic features
of helicopters.
1. Development of Helicopters in Size
A study of foreign helicopters indicates that the use of helicopters for
landing Marines from ships was the determining factor in the further development
of military helicopters as troop carriers. The American landing of troops in
S-55 helicopters at Inchon during the Korean War (1951) was a typical exanple
of this trend.
The size range of the assault helicopters was predicated on bulk and weight
of ground transportation means used by the troops and to be dropped by air. It
is a known fact that conventional weapons - mainly artillery - transported by
prime movers are close in weight to the weight of the prime movers themselves.
Thus, the lift capacity of the first transport helicopters in annies of other
countries was 1200 - I6OO kg (the weight of a light military truck used as
prime mover together with the re^ective weapons). Subsequently, the required
lift capacity of helicopters was increased to 6 - 8 tons which, in accordance
with military technique, was based on automobile carriers with a lift capacity
of 3 - 4 tons. Still later, for exanple in projects developed by Sikorsky Air-
craft, the lift capacity of helicopters rose to 20 - 25 tons and finally to
36 - 40 tons. Such weights correspond to the weight of light and mediim tanks
or of self-propelled landing craft. Whether .this development trend in size inr-
crease will ever come to an end depends on the constantly changing military
planning. Artillery systems are being largely replaced by missiles, for which
reason the foreign press often mentions the need to transport missiles or missile
systems, the prime factor in determining the size of modern helicopters.
In the atteiifjt to single out the main trend of future helicopter develop-
ment, after successively outlining the creation of new types of machines in the
few designer firms that have been successful in developing experimental models
into practical prototypes and in starting pilot series, it will be found that
the major development was toward an increase in the lift capacity of helicopters.
TABLE 1.1
Characteristics
USSP
Mi-1
Mi -4
Year of production
1948
1952
Lift capacity, in
ton- force
0.3
1.2—1.6
Increase over previous
model
4
Flying weight,
2.3
7.2
ton- force
Heli copters
1957
8—12
39—41
USA
S-51 S-58 S-64 DS-103
1946
1956
1962
Projects
0.3
1.2
5-6
20
3
4
3
2
6
17.0
—
40
2
Table 1.1 gives data characterizing the development of the lift capacity
of single-rotor helicopters of the same configuration by two aircraft construc-
tion departments - helicopters Mi-1 (Fig. 1.2), Mi-4 (Fig.l.3), Mi-6 (Fig. 1.4),
Mi-10 (Fig.1.5), S-51 (Fig. 1.6), S-58 (Flg.1.7), and S-64 (Fig. 1.8).
As we see from Table 1.1, the lift capacity increases severalfold with
each prototype.
However, it is easy to show that an increase in size and weight of heli-
copters is impossible without a qualitative inprovement of the engines used /lO
(reduction in weight per unit horsepower and increase in econony, i.e., decrease
in fuel consunption) .
Actually, an increase in flying weight is possible either by increasing the
rotor span or the installed power, or both factors
G = T=(k^ND)'l'.
(1.1)
The weight of the engine is proportional to the first power of its output,
while the weight of the machine itself increases only in proportion to the 2/3
power .
Fig. 1.2 Mi-1 Helicopter.
Thus, a helicopter with a larger power-to-weight ratio will have a rela-
tively greater design weight, owing to the power plant.
In like manner the weight of the blade and, accordingly, the weight of the
lifting system change in proportion to the third power of the diameter, whereas
the weight of the helicopter again changes only in proportion to the 2/3 power.
Here also, the weight of the lifting system of a larger helicopter proves to be
relatively greater. Thus, on increasing the size of a helicopter its load
ratio, i.e., the ratio of useful load to flying weight, should be decreased, if
there is no weight inprovement in engines, blade design, reduction gears, or
transmissions. Actually, in the 1930s papers were published that demonstrated
the uselessness of developing helicopters with a power greater than 500 tp,
since an increase in power would not lead to an increase in usefiol load. /13
According to technical specifications of that time, the weight of rotors, reduc-
tion gears, and of the entire machine as a whole increased with increasing power
more rapidly than the lift.
However, in developing a new militaiy - and especially a new general-
purpose - helicopter, the designer will not tolerate a lowering of the achieved
level of load ratio.
Thus, a "quantitative" development with respect to size is inpossible with-
out a qualitative development; in fact, it always is concurrent with the qualita-
tive advance of technology.
The development of helicopters larger than the first two- or three-place
models took place in a conparatively short time, since the unit weight of piston
engines always decreased with an increase in power. But in 1953, after develop-
ment of the 13-ton Sikorsky S-56 helicopter (Fig.l.lO) with two 2300- tp piston
m
Fig. 1.3 Mi-4 Helicopter.
Fig. 1.4 Mi-6 Helicopter.
Fig. 1.5 Mi- 10 Helicopter.
m.
Fig. 1.6 S-51 Helicopter.
Fig. 1.7 S-58 Helicopter.
Il.g.1.8 S-64 Helicopter.
^takeoff 'on-force
40
'm ^"«i^AM«ki^B -
engines, the size series of helicopters in the West was discontinued and only
in the USSR was it possible, in 1957, to develop the Mi-6 helicopter with a
flying weight of 40 tons by using turboprop engines.
2. Qualitative Development of Helicopters
In the middle of the 1950s, the reliability of helicopters became appreci-
ably greater so that also their use potentialities for the national econon^r in-
creased. This moved problems of econong^ into the foreground.
The operating cost per hour of a helicopter plays a decisive role in
whether to use them for geological surveys, in agricultxore, or for transporting
passengers. Amortization, i.e., the price of a helicopter divided by its
service life, constitutes a large portion of the cost. The service life of the
helicopter is determined by the durability of its conponents. The problem of
increasing the fatigue strength of blades, shafts, transmissions, rotor hubs,
and other lonits of the helicopter became a prime problem, which helicopter de-
signers are still studying at
present. Today, a life of 1000
hours is no longer a rarity for
series-produced helicopters and
there are no grounds to doubt its
further increase. When using
helicopters in transportation,
the concepts of cost per ton-irdle.
of the transported load and the
cost per passenger-mile become
decisive. This is the hourly op-
erating cost divided by hourly
productivity, i.e., by the product
of the weight of the pay load and
the cruising speed.
Since the construction weight
largely determines the price of a
helicopter, the direct relation
between econon^r and load ratio of
the helicopter is obvious. Flying
speed also acquires a new role.
/m 1950 IS55 I960 I9B5
This automatically leads to
Fig. 1.9 Size Evolution of Helicopters. the idea of developing helicopters
with higher economic indexes. The
development of turboprop engines
with an appreciably smaller unit weight than piston engines made it possible to
produce helicopters with a larger load ratio while retaining, in each weight
category, the rotor dimensions.
Generally, replacement of piston engines by turboprop engines not only
results in a decrease in relative weight of the power plant but also in some
increase in power; produces a dual effect and also leads to an appreciable
^^^5l3r^'
i
S
increase in criiising speeds.
In the diagram (Fig.1.9) we traced these quantitative and qualitative de-
velopment trends of the most common helicopters produced by the three design
engineering departments. Given are the single- rotor helicopters designed by
SikorslQT- Aircraft (USA), the single-rotor Soviet helicopters, and the fore-
and-aft helicopters of the Piasecki Aircraft Corporation, which subsequently
became the Vertol Div. of Boeing.
I2k
Fig. 1.10 S-56 Helicopter.
Thus, the size development trend on the basis of piston engines (soUd
lines in Fig.l.9) was terminated as early as 1953. Then, as turboprop engines
of the necessary size were developed over a period of five to ten years, second-
generation helicopters appeared (points referring to these in the diagrams are
connected with the original models by the broken line of qualitative develop-
ment).
Thus, the helicopters S-55, S-58, and S-56 with piston engines served /16
as prototypes, respectively, for the turboprop machines S-6I (Fig. 1.11), S-62
and S-65 (Fig. 1.12;. The same holds for the fore-and-aft helicopters of the
Vertol Div. of Boeing V-107 and YBr-Hk "Chinook" (Fig.1.13).
The Soviet turboprop helicopters Mi-2 (Fig. 1.14) and Mi-8 (Fig.1.15) also
constitute a ftirther development of the well-known helicopters Mi-1 and Mi-4.
The unusually long service life of helicopters is striking in conparison
with airplanes. Almost all piston helicopters shown in the diagram (with the
exception of the e^qjerimental helicopters XH-16 and S-56) were in production and
service before the appearance of their second turboprop generation, and the Mi-1
helicopter has managed to stay in production for 15 years and is approaching
the record longevity of the Li-2 airplane.
We can assume that the weight categories of helicopters indicated in
|[
1 *^*s,^''%>^;-!:;
m.
Fig. 1.11 S-61 Helicopter.
Jig. 1.12 S-65 Helicopter.
Fig. 1.13 Chinook Helicopter.
10
Table 1.2 have 'beconie established hy now.
What -will be the future development of helicopters?
/M
The process of developing a new generation of helicopters, on the basis of
inp roved turboprop engines, is now being conpleted in the lightest category of
helicopters. The lag in this weight category can be attributed to difficulties
Fig, 1.14 Mi-2 HelLcqpter.
Jig. 1.15 Mi-8 Helicopter.
in developing a lighter and simultaneously more economic low-power t^lrboprop
engine in conparison with piston engines. In the end, such an engine was de-
veloped in the USA by the Allison Company - this was the T-63 weighing only
174 lbs at a power of 3I5 hp and a consunption of 280 gtn/hp-hr. The award in
the conpetition for a light three- or four-place military helicopter in the USA
was made to the Hughes Aircraft Conpargr, which created the UH-6A helicopter
(Pig. 1.16) weighing only 2680 lbs at an enpty weight of about 134O lbs; this is
11
m.
Fig. 1.16 Hughes Helicopter UH-6A.
Fig. 1.17 Fairchild HLller Helicopter FH-1100.
an appreciable technical achievement which required a number of new design solu-
tions, in particular the use of a rotor with an elastic spring retention of the
blades instead of the conventional hinge attachment. This helicopter has a high
load ratio {50%) combined with a high cridsing speed (213 km/hr), for a light
machine. The Fairchild HLller FH-llOO is also in this class (Fig. 1.1?). It is
obvious that these helicopters considerably outstrip the light Haison recon-
naissance aircraft of World War II, both with respect to speed and lift capacity
and, fiirthermore, have the great advantage of vertical takeoff and landing.
Thus, the decision made in a number of countries to replace light reconnaissance
aircraft by helicopters is not surprising.
12
TABLE 1.2
Ch
aracteri sties
Lift capacity or
number of places,
Flight weight
Light
!.i ai son
2 — 4 per-
sons
1.5—2 ton
Light
Multi-
purpose
Type of Helicopter
Light
Transport
1 ton or 3 ton or
10—12 per- 25—30 per-
3.5 — 4 ton
10—12 ton
Medium
Transport
Heavy
Transport
Superheavy
Transport
6 — 8 ton
20— 40 ton
20 ton
40 ton
Of course, a new generation of light helicopters will also be developed in
other countries of the world. In France, this is being done on the basis of the
350- Ip Turbomecca-Oredon-III engine. In West Germany, the Bolkow ConparQr is
working on such a machine.
Thus, in speaking of the qualitative development trend of helicopters, it
is obvious from the foregoing that each new generation of engines gives rise to
a new generation of helicopters in all weight categories, simultaneously having
greater econonQr and better flight performance data. This line of development
probably has no upper limit.
As regards the size evolution of helicopters, no machine with a lift capaci-
ty of 20 tons (see Table 1.2) has been developed as yet.
According to a request for proposals, announced in the USA, firms such as
Kaman, Fairchild Killer, and Sikorsky Aircraft are working on the development of
a helicopter with a lift capacity of 20 tons. In West Germany, the Bolkow
Conpany is working on a helicopter with a 40-ton lift capacity. Below, we will
review the possible ways of developing heavy and siperheavy helicopters.
3« Speci al-Purpose Helicopters /19
It is necessary to mention also the development of various models of
special-purpose helicopters within the indicated weight categories. In this
connection, let us make a brief remark on the new concept of using helicopters
in the Army which has recently developed in the West - especially in the USA -
namely, the creation of so-called airborne mobile troops.
In this instance, helicopters are used in place of motorized transport for
all types of troop movement. The Bell "Iroquois" helicopter UH-ID (Fig. 1.18) is
particiJ.arly adapted for transporting troops by platoons (11-12 men).
light reconnaissance three- or foua>-place armed helicopters (Hughes heli-
copters 0H-6a), flying in front of battle formations, are also a necessity.
Finally, regular troop-carrier helicopters of various classes, supplying the
13
means of gro-und fire sipport such as artillery, rockets, and tanks, take over
the task of troop movements.
Also used in realization of this concept are helicopters for air sipport of
infantry, constituting a vinique type of assault helicopters. Ordinary heli-
copters armed with radio-controlled missiles and weapons are presently used for
this purpose.
Fig. 1.18 Bell "Iroquois" Helicopter UH-ID.
Such an airborne motile division is supplied from the air by airplanes and
helicopters of the Mr Force Materiel Command.
It is not difficult to detect behind this concept past military experience,
wherein any new type of transportation that became accessible engendered a new
type of troops. Beginning with cavalry, we recall the bicycle and motorcycle
xinits of World War I, and the motorized infantry, motorized divisions, and air-
borne troops of World War II.
It is clear by now that this concept is finding followers in many western
coiintries*
Thus, the l2-place SA-330 (Fig. 1. 19) helicopter ordered by the French Army
corresponds to the 11-place Iroquois helicopter (USA). A similar machine is
being designed also in West Germany.
The need to retain the class of 10- to l2-place light transport helicopters
is confirmed also by the practical esq^erience with the l2-place Mi-Zt- helicopters
in the national economy. It is obvious that the development of more economic
(for airlines) 30-place Mi -8 helicopters does not interfere with the advantage
of using the 10-place helicopters in the national economy for geology and other
purposes.
14
4« Compound Helicc^ters -with Additional Engines - Rptocraft /20
Of considerable interest was the appearance of conpound helicopters which
use propellers for forward flight, as autogiros did earlier. Such are the
Rotodyne Ferry designed by Hislqp and especially the rotocraft of the Soviet
designer M.I.Kamov.
In 1964, world records for machines of this type were set on the rotocraft
Ka-22: speed 36O km/hr, Hft capacity 16 tons. N.I.Kamov's rotocraft again
focused attention of the helicopter world, after 20 years, on the side-by-side
configuration which had been successf uUy developed by Focke in Germany and by
I.P.Bratiikhin in the USSR. This machine recalled the great advantages of the
side-by-side configuration in flying range and lift capacity with a running
takeoff which must be accoTinted for in a successful design.
'**fe£
(.. mf''^"'"' ^'^^^
^•^•^Nsi,
Fig. 1.19 SA-330 Helicopter.
A further development of conpound helicopters with propeller is represented
by the helicopter prototype with additional turbojet engines now being proposed
in the West for military purposes.
An interesting prototype of an assault helicopter is the Lockheed composite
helicopter (Fig. 1.20;. This two-place experimental machine, in addition to the
main 550-lnp turboshaft engine driving a four-blade rotor with elastic blade re-
tention, uses a turbofan engine mounted on a small wing and permitting rev-up
to 426 km/hr when briefly cut in during flight .
The successful development of dual-flow turbofan engines, especially with
a large bypass ratio, may lead to the development of models which, at cruising
speed, would have a specific consimption of the order of Cr = 0.5 kg/kg ♦ hr.
Since
^ _ 75r,
V
■ C
/?•
it is not difficult to calculate that, in this case, the consunption per horse-
power of an equivalent propeller engine at a propeller efficiency of 0.75 and a
flying speed of I50 m/sec is only about 200 ^n/(lp - hr) .
15
If we also take into accoiuit the small weight of such a motor in conparison
with the weight of a tTH-boprop engine, it becomes clear that the use of turbofan
engines of this type can be economically ad-vantageous even at lower cruising /2l
speeds and may lead to the development of conpound helicopters with an auxiliary
thrust engine and wing for passenger transport between urban centers at cruising
speeds of the order of 350 - 450 km/hr. At the same time, such helicopters may
find military use as fire-stpport craft for troops.
Fig. 1.20 Lockheed Helicopter XI^51A.
In analyzing the ways and means of helicopter development, one cannot side-
step the question of vertical takeoff aircraft. Will the development trend and
use of helicopters come to an abrupt end with the appearance of such craft, as
had been the case with autogiros when helicopters came into being?
Section 2. The Helicopter Co mp ared to Ve rtical Takeoff
and Landing and Short Takeoff and Landing
Aircraft
When talking of the prospects of helicopter engineering development, one
must study the problem of the possibility of coexistence of helicopter and
vertical takeoff aircraft. Do helicopters have a future? Or are the potenti-
alities of the helicopter exhausted? Can the helicopter successfully conpete
with vertical takeoff aircraft? Will their development trend teniiinate, as was
the case with autogiros which ceased to exist with the appearance, in 1940, of
the first successful helicopters? A conparative investigation of helicopters
and VTOL or STOL craft as means in transport aviation not requiring an airfield
will enable us to answer these fundamental problems.
It is known that recently the matter of vertical takeoff aircraft (in
English, VTOL) and short-rion aircraft (in English, STOL) has become urgent"'.
(For footnote, see following page).
16
Actually, the present flying speed of fighters, reaching 2500 - 3000 km/hr,
requires such high-power engines that very little remains to add for their
vertical takeoff. Therefore, judging by the literature in other countries we
can assume that fighters and fighter 'bomtiers will be developed mainly as /22
VTOL aircraft not requiring the use of an airfield. The direction of develop-
ment of transport aircraft, whose power plant is limited by considerations of
economics or quite sinply by fuel consumption, tends toward STOL aircraft .
Some propose that the future development of helicopters will offer a better
solution to transport problems for a range vp to 600 km than do VTOL aircraft
or special STOL transport aircraft.
In examining the possible development trend of aviation, we cannot limit
the study to an analysis based on the present state of the art in science and
technology.
By using such methods, many scientists have repeatedly arrived at erroneous
conclusions concerning the "limits" in the development of various aircraft or
helicopters, since they did not provide for the development of parameters' char-
acterizing the weight and economic perfection of engines or perfection of design
and materials used. It is necessary to extrapolate their development somehow to
the future.
Leaving room in the future for such an investigation, we will estimate the
situation at hand. We will conpare helicopters with VTOL and STOL aircraft,
using data of the best helicopters that have been bixLlt as well as of aircraft
being in the design or construction stage.
1. Tactica l an d Technica l Require ments for VTOL, and STOL
Military. Transport^ Ai r craft of the West
The tactical and technical specifications for VTOL transport aircraft,
worked out in the USA, call for a flying range of 550 - 700 km, a lift capacity'
of 3600 kg or 32 troops, and a cruising speed of l+^Q - 550 km/hr at a gross
weight of not more than 16,000 kg. At the same time a very long delivery range,
of the order of 4000 km, is required, which is probably intended for the possi-
bility of ferrying aircraft from the USA over the ocean.
In studying STOL transport aircraft, one comes across ordinary classical
propeller transport planes such as, for exanple, the British-Canadian De Havil-
land "Caribou" (K-g.l.2l).
By STOL transport aircraft we mean aircraft that use engine power for /23
reducing the takeoff and landing runs. This is useful and necessary.
A study of STOL aircraft must include one of the first aircraft of this
type, the French aircraft Breguet-941 (Fig. 1.22). On this aircraft the entire
wing area is in the zone of propeller slipstream. All propellers are inter-
■5!- VTOL - vertical takeoff and landing; STOL - short takeoff and landing.
17
connected by a transmission which provides safe takeoff or landing if one or
two of its four engines fail. The propeller slipstream, deflected downward by
a double-slotted flap, produces additional lift, which reduces takeoff speed
and shortens the rtin. However, these qualities are achieved at the e^gjense of
an increase in enpty weight and shorten the range of this STOL aircraft. Heli-
copters can operate successfully at such a range.
Fig. 1.21 British-Canadian Transport Plane
De Havilland "Caribou".
Fig. 1.22 French STOL Aircraft Breguet-9/(-l .
Despite the great type diversity of ?TOL and STOL aircraft, it is not dif-
ficult to arrange them logically in a general classification of aircraft . They
should be placed between helicopters and airplanes.
It is commonly known that the larger the area over which air flows (it
makes no difference whether it flows through a rotor or the nozzle of a jet
engine) or, more precisely, the smaller the velocity inparted to the air mass
for producing lift in aircraft or helicopter, the smaller will be the power re-
quired for this per unit weight of machine.
Thus, the ordinary helicopter and the aircraft taking off vertically by the
thrust of jet engines are at opposite poles of this classification (Fig. 1.23).
18
In the pursirLt of greater range and probably higher speed, the helicopter
was provided with a wing; as the wing area and hence the lift increased further
(since the thrust of the rotor at maximum flying speed decreases so much that it
is insufficient for foiTward flight), propellers appeared on the wing. Thus arose
the British "Rotodyne" (Fig. 1.24) and the Soviet rotocraft designed by N.I.Kamov
(Fig. 1.25) - aircraft which in place of one lifting and moving system have two,
one being the rotor and wing for sustention and the other being a system of
tractor propellers, inclined forward to the thrust vector of the rotor, to pro-
vide foirward propulsion. During vertical takeoff, the wing and the propellers
are useless, and in horizontal flight the rotor is siperfluous. The attenpt to
avoid such superfluous units whose weight unavoidably reduces the useful load
led to a configuration with a wing and pivoted rotor (Bell XV-3, Fig. 1.26) in /24
which the rotor in horizontal flight becomes a propeller, and to a configuration
with a pivoted wing whose propellers during takeoff - turning together with the
wing - act as rotors as, for exatiple, the XG-142 aircraft produced by Chance
Vought - I^an - HLller (Fig. 1.27)-
Jet aircraft taking ofi
on special engines
VTOL aircraft
Propeller aircraft Jet aircraft taking
with pivoted wing off on fans
(Miller ^C-i42) '
tiotocraft "Rotodyne'*
Aircraft with short run
of 130-160 m
Mi-6 h
Light transport aircrafti^(Bregiiet-9^1)
with run of 16ri-:?nn m
(Car ibou)
anspor t
a run
of 600-800 m (An-10)
Fig. 1.23 Classification Scheme for VTOL and STOL Aircraft.
Fig. 1.24 Rotodyne Rotocraft.
19
Passing now to aircraft with an engine more powerfiJ. than that of the above
types of aircraft, the STOL jet aircraft is provided with means for downward de-
flection of the blast from the jet engines or from various types of auxiliary
turbofan engines.
The configuration of the Breguet-941 aircraft (see Fig. 1.22) can be re-
garded as a variant of an ordinary airplane which, to increase the lift coeffi-
cient, utilizes the airflow over the wing created by the propellers, or else as
a variant of an aircraft with a pivoted wing where the thrust of the propellers
is not literally turned but is deflected downward by means of the mechanized
wing.
Fig. 1.25 Rotocraft Designed by N.I.Kamov.
Z2i
Fig. 1.26 Bell X7-3 Convertiplane.
The diameter of the propellers of the WOL aircraft shown in Fig. 1.23
(from left to right) gradually decreases down to the VTOL jet aircraft which
20
II
has no propeller at all. With a reduction in propeller diameter, the engine
power increases per unit takeoff weight from 0.25 - 0.3 IpAg for helicopters
to 3 - 4 1^/kg for jet aircraft (the values of the equivalent horsepower are
taken here for the aircraft ) .
Z26
The cruising speed of these aircraft continuously increases along with the
increase in installed horsepower. However, this is not a decisive factor for
the problem of a transport aircraft with a range of 800 - 1000 km.
This defines the scope of VTOL and STOL transport aircraft to be conpared
and the flying range over which they are effective.
Fig. 1.27 Chance Vought - %an - Hiller XC-1^2 VTOL
Aircraft with Tilt Wing.
To which of these types of aircraft will belong the future in solving the
formulated problem?
Before conparing the helicopter with its competitors with respect to econo-
my, let us examine the problem of the flying range of the helicopter. In view
of its conparatively short range, can the helicopter enter this conpetition at
all?
Let us first examine and conpare the best of the VTOL and STOL transport
aircraft that have been or are being constructed: the tilt-wing VTOL air-
craft of the type XC-142; the STOL aircraft of the type Breguet-941; the regtilar
transport aircraft of the type "Caribou" HG-4; the rotocraft with turboprop
engines of the "Rotodyne" type; and helicopters.
2. Mea ns for Increasing the Flying Range of Helicopters
The helicopter. has always been regarded as a short-range aircraft; a figure
of 400 - 500 km is usually given as the maximiim for its normal range. In order
to treat the helicopter as a conpetitive aircraft in this new area of use, the
21
range should be almost doubled while retaining its lift capacity. How does one
increase the flying range?
Let us turn to the well-known formula of flying range:
1 = 270-^^-^
G c^ C/
121
(2.1)
where
Ce
G = weight of the aircraft (average during flying time);
Gt = weight of the fuel;
= aerodynamic efficiency of the aircraft (taken to be constant);
= specific fuel consimption of the engine;
= a coefficient taking into account power losses in the transmission
due to cooling, etc.;
Tl = rotor efficiency.
Equation (2.1) shows that the range is greater, the larger the proportion
of fuel in the all-up weight of the aircraft and the higher its aerodynamic ef-
ficiency, engine economy, and effi-
ciency of engine and auxiliary units .
This formula holds for any
heavier-than-air craft, including air-
planes and helicopters. Specifically,
it follows from this equation that the
flying range of various flying
machines, other conditions being
equal, does not depend on their cruis-
ing speed.
'■jtri
I
1
Airpl
-r
ane
-=j-
15
5.0
>J
-^'^
-^
\,
/
^
Hel icop ter
N
2.5
^
'
so 100 t50 ZOO 250 300 f-^
Can a helicopter be given a range
sizfficient for coizpeting with STOL
aircraft?
Fig.l.2S Product of Aerodynamic
Efficiency and Rotor Efficiency
as a Function of Flying Speed.
As indicated in Fig. 1.28, the
product of aerodynamic efficiency
Cy/cx and rotor efficiency 7] for a helicopter with a fixed landing gear is lower
than for a transport airplane by almost a factor of 2. Furthermore, the fuel
consunption of the helicopter is somewhat greater than that of the airplane
since the engine characteristics are inferior at low altitudes and flying speeds.
Thus, a helicopter can be given a range equal to that of airplanes only by in-
creasing the fuel svp-plj, i.e., the quantity G^/G. However, in so doing how
does one maintain the useful load? This can be done only by increasing the
takeoff weight, but the helicopter will then no longer be able to take off
vertically.
What happens if we place these aircraft under equal conditions, i.e., allow
the helicopter the same takeoff run as an STOL aircraft, namely 150 - 200 m or
even less? At a relatively large value of Cy, will the helicopter then be able
to lift - at low speed - a much greater weight than an airplane, accommodate
22
more fuel, and thus condensate for its lack in aerodynamic efficiency?
3. Helicopter with Takeoff Run
As shown in Fig. 1.29 which gives the curves of the required and available
horsepower of a "Cari'bou"-type transport airplane and of a modern helicopter, an
airplane can be kept in the air at a speed not below 115 km/hr. A helicopter /28
can hover in the air without moving. If
the heUcqpter is overloaded by 15% above
the nonnal takeoff weight Gq, it can no
longer hover and, like the airplane, will
only be able to fly without dropping if it
has some speed - in this case, a speed of
not less than 50 km/hr. At a greater
speed than this, it will gain altitude and
at a lower speed, lose altitude. The dif-
ference here in favor of the helicopter,
in conparison with the conventional air^
plane, lies only in the fact that the
helicopter retains full controllability
at a speed below its minimal and that
there is no danger of separation of flow
and loss of controllability, both of which
are possible in the aiiplane.
»^p
3000
zooo
1000
Required horsepou
for airplane
wo
200
300
Fig. 1.29 Required and Available
Power as a Ftinction of Flying
Speed.
So far as the takeoff distance is
concerned, assuming that the helicopter
takes off at a speed of V^in, this dis-
tance at some average acceleration j,
will be
Z.„,
2
(2.2)
Thus, the takeoff vxm is shorter, the lower the minimum flying speed (close
to takeoff speed) and the greater the acceleration.
The minimum speed is
V„
V °^- S,nax
where
wing area;
air density.
What values of c.
are available to airplanes and helicopters?
For this, let us calculate the value of Cy that an airplane of the "Caribou"
type should have at the same weight as the helicopter in order to fly without
descending at speeds less than minimum. Figure I.30 shows the values of Cy,
23
calculated from the formiila
c.,=
20
eSwK2
(2.3)
of a helicopter referred to the wing area of an equivalent airplane, which char-
acterizes the lift capacity of a helicopter in conparison with the airplane.
The cin-ve Cy of the helicopter in Fig. 1. 30 extends to infinity. This is natural
since the helicopter has a rotor which in essence is a rotating wing with a
power plant suspended from it and is capable of producing lift at zero forward
speed of the entire machine. Here we see that at speeds of 50 - 60 km/hr the
available values of Cy of the helicopter are several times greater than for an
airplane of the "Caribou" type at a speed of 115 km/hr, which has a highly
mechanized wing.
Thus, at equal power a greater weight can be lifted by the helicopter /29
at low speeds than by an airplane.
However, a greater flying weight
does not always mean a greater useful
load.
At equal relative fuel weight
(about 10^), the ordinaiy airplane of
the "Caribou" type has a range of
1000 km, i.e., twice that of a heli-
ccpter taking off without a run.
The Breguet-941 STOL aircraft (at
a fuel weight 12 - 13^ o'f the flying
weight) has twice the range of the
helicopter or of the X6-142 VTOL air-
craft .
If, in helicopters, the fuel weight
is increased to 20 ,- 25^ of the gross
weight, then the range of the helicopter
can be doubled and raised to 1000 km.
This value is already close to the
normal ranges of specially designed
STOL aircraft.
(■''he
1
10.0
8.0
BJO
\
U.0
(<■
\
\
^
2.0
\
1
\
^l^'lLin
,1 .
aire
Biin
^
(G=WGg] 100
200
hr
Fig. 1.30
Dependence of (cy ^j),
on Flying Speed.
sq
The load ratio of helicopters
taking off with a imn and at increased
fuel sT^jply becomes higher than the load
ratio of coiiparable aircraft and reaches
44 - 50^. This makes it possible 'to ob-
tain equal productivity at almost the same takeoff weight of airplane and heli-
copter. For exanple, a transport helicopter of average lift capacity, just as
a "Caribou"-type airplane, can transport a load of 3.2 tons over a range of
1000 km. It is true that the helicopter, in so doing, uses 2.5 times more fuel.
However, it must be remembered that the airplane needs twice the area for taking
24
off and, what is quite inportant, the helicopter after having constimed half its
fuel is able to land vertically, whereas the ordinary aiiplane cannot do so.
It must be enphasized that conparable aiiplanes and helicopters have
practically the same power supply (0.23 - 0.25 Ip/kg). One must also bear in
mind that piston engines, operating on gasoline, have a lower fuel consunption
at low altitudes than turboprops, so that the average tvirboprop helicopter oper-
ates under less advantageous conditions than the "Caribou" aircraft with piston
engines .
Thus, the suggestion to use a takeoff run for the helicopter will permit
doubling its range at the same useful load.
4« Takeo ff Distance of Helicopter
We have already expressed the takeoff distance in terms of takeoff speed
and acceleration. The takeoff speed, proportional to the minimum speed at which
a helicopter can be supported in the air at an overload of 15^ as opposed to /30
the weight with which it can take off
without a run, is not more than
"" '"'"-- - 60-70 km/hr. Let us now define the
possible degree of linear accelera-
tion, since the takeoff run is in-
versely proportional to acceleration.
Let us find the possible initial ac-
celeration.
T«-T*T
nose wheel
tail wheel
As agreed, let the helicopter
develop a thrust amounting to only
0.85 (takeoff weight) at the take-
off power. Then, allowing for some
angle of inclination of the rotor
axis to the vertical Sf (here the
difference in the compression of the
struts and pneumatic tires of the
nose and tail wheels is accounted for) and for the forward deviation of the re-
sultant owing to deflection of the automatic pitch control mechanism through an
angle Dik, according to Fig. 1. 31, we find the initial acceleration:
Fig. 1.31 Forces Acting on Helicopter
during Takeoff Run.
^0= 5-[-^sin(D,x-;-£^)-^l--^yj,
(2.4)
Here, the second term on the right-hand side takes into accoiont friction
of the wheels against the ground, with a friction coefficient f . Adopting the
T
usual notations of e, = 6.5°, DiH = 10°, -^ = 0.85, and f = 0.12, we obtain
Jo = 2.2 m/sec^.
p
Assuming a relative static propeller thrust of -rj- = 1.6 kg/tp for the
25
airplane and us5 .g = 0.25 i^/'i^gf the initial acceleration -will be
G
/n=fi'( /W2.5 m/sec^
Jo^g
\N G
i.e., a value of the same order as for the helicopter.
Of course, the acceleration at the moment of takeoff is determined by the
excess power, which is somewhat higher for the airplane. However, its propeller
thrust decreases with an increase in speed whereas the rotor thrust increases;
in fact, the angle of pitch of the
heUcqpter, during the takeoff
run, may even increase since,
during takeoff, the tail wheels
are able to Hft off the ground
at a thrust substantially less
than the takeoff weight so that
the takeoff run is conpleted- on
the nose wheel.
Fig. 1.32 Forces Acting on a Helicopter
during Takeoff on Nose Wheel.
(-0)
-1
takeoff
It is obvious from Fig. 1.32
that the thrust-to-weight ratio at
which the tail wheels can lift off
the ground (disregarding friction)
will be
(2.5)
where x„ax is the distance from the center of gravity to the axis of the auto-
matic pitch control mechanism at a maximal forward deflection/ of this mechanism
[here it is assumed that the quantity
x„
can be neglected for unity] .
Fig. 1.33 Running Takeoff of Mi-6 Helicopter.
26
With the usual relations, this corresponds to a thrust-to-weight ratio of
0.8 - 0.85 • Figure 1.33 shows a helicopter during the takeoff run, at a gross
weight of G„ax = 1.15 &•
An exact calculation of the takeoff run can be carried out by the same /31
method proposed by the author 30 years ago for calculating the takeoff run of an
autogiro ( Ref .4) •
Running takeoffs perfonned in practice have confirmed that, at a 15^ over-
load of a helicopter as opposed to the maximal weight with which it can take off
without a run, the takeoff run amounts to no more than 60 - 100 m in still air.
5. Criterion for Estimating the Economy of Various
Transport Aircraft
In any conparison of two transport aircraft, attention is primarily centered
on the lift capacity. Still, the speed of transport is also important . Actual-
ly, if a load can be transported more quickly, then more loads can be trans-
ported in unit time over a given distance at a smaller lift capacity.
This results in the well-known criterion of hourly productivity Gio^d^av "t '
• km./hr (Y^^ is the average ground speed).
However, at what cost is the load transported?
If both aircraft have identical efficiency and range as well as takeoff /32
and landing properties satisfactory for fulfilling the mission, which should be
given preference?
To answer this question we must know which of the aircraft is more economi-
cal. In military use, the advantages of any aircraft for solving transport
problems, which sometimes arise at an appreciable distance from the supply
bases, are determined primarily by cost data. Expenditures for construction of
the machine itself, incurred in the past, are no longer of significance and have
no effect on fulfilling the immediate task. Under such conditions, the economy
of an aircraft is determined mainly by the amount of fuel consumed. Here, the
transport of fuel constitutes a bottleneck that is decisive for the ability to
solve the stated problems . The criterion of economy under such conditions is
conveniently obtained by referring the hourly productivity to the weight of the
fuel consumed during that time Gtiji:
^..= ,^. (2.6)
Since the fuel cons^l^ption per kilometer is
^ (2.7)
it follows that
27
The quantity L^^g has the dimension of length, so that we can call it the
equivalent specific range of the aircraft. It represents the distance over
which a given aircraft can fly in excess of the design range if the entire trans-
ported load is replaced by fuel. Still another meaning can be given to this
quantity. It can be regarded as the distance over which an aircraft can carry
one ton of cargo after having consumed one ton of fuel. It is clear that the
quantity L^.g depends on the distance of transportation just as productivity
depends on it. The farther the machine flies, the more fuel it needs and the
smaller the cargo it can take at a given flying weight (maximal).
On the other hand, L^^g is the work expressed in ton-miles which a given
aircraft can perform, having consumed one ton of fuel.
The inverse quantity of L^ ^ , i.e., — = , is fuel consunption in tons
required for performing transportation work of 1 ton-mile or kg/ton-mile.
We can also use other criteria that estimate economy, i.e., the cost of
transporting one ton-mile in rubles. In these criteria, we can take into ac-
co\ant the cost of the aircraft (to some extent, this is proportional to the
entity weight of the aircraft), the service life of the conponents and power
system, cost of operation and repair, etc.
In this problem, the military transport criterion L^^ ^ or — — - — = Cl is
most iirportant since it takes into account not only econony but also the real
and ponderable requirement of sijpplying fuel for transportation equipment under
military conditions. Therefore, we can disregard roany other criteria but not
this one.
We sipplemented the values of L^^g and Cl for conparable aircraft by /33
data on jet VTOL aircraft which, for hovering, use specially installed lifting
jet engines or which take off by means of fans driven by the main engines. The
fuel consunption of such aircraft, while hovering, constitutes such a large per-
centage of the takeoff weight that it must be taken into account when calculating
the value of Cl ; therefore, the corresponding formula takes the form
where iqOO G,
Cl • = -p j^— is the fuel consunption xn horizontal flight;
"h O V
Cl = 1000 — ■- is the fuel consumption while hovering.
G
For an aircraft taking off by means of wing fans rotated by turbines
mounted to their periphery, we have used data similar to those published for the
experimental Ifyan X-16 airplane constructed in the USA.
In the calculations, we disregarded the fact that the installation of the
28
lifting engines inpairs the aerodynamic efficienciy of a VTOL aircraft. It was
assumed that, at ranges greater than 1000 km, VTOL aircraft of the indicated two
types with a sustainer turbojet will fly at the design altitude at maximum aero-
dynamic efficiency, whereas at a reduction in flying range from 1000 to 50 km
the operational ceiling decreases accordingly and any drop in aerodynamic effi-
ciency leads to some increase in fuel consuoption. The hovering time t^ov was
taken as 6 min (3 min in takeoff and 3 niin in landing) .
Table 1.3 gives owe calculated data for aircraft of different flying ranges
and different forms of takeoff.
TABLE
1.3
Vertical
Typ e o f Tak ec
With Short
Run
i, <200 m
ff
From Small
Airfield
I, <400 m
Fr";
Con
Air
Characteri sties
)m
:rete
strip
«8D0m
Road
Transporta
Range t,. km
■ 500
1000
1000
2000
_
Flying altitude
V
a.
o
u
u
X!
500-2000
500-2U0O
3000
8000
-
Typ e o f
fli rcraft
VTOL
Propell er
' Aircraft
4->
(0
U
o
o.
o
o
■ H
STOL
Propeller
Aircraft
CO
t-<
o
2
£
Light Transport
Propeller
1 Aircraft
1 .STOL
Propeller
Aircraft
(Overload Weieht:
[ Turboprop
Transport
Aircraft
Turbojet
Transport
Aircraft
° s
[- a:
a
a
I 3
m u
«i-
o
u
o
Takeoff run
I , m
80
140
1£0
170
210
850
950
—
—
't'
C^, kg/tkm
0.8
0.9
1.3
0.9
0.6
1.4
0.4
0.5
0.3
0.5
0.13
0.4
V^^, km/hr
250
460
280
240
400
280
250
400
600
800
60
30
To conpare the economy of transport conveyances, the values of Cu are given
for the Soviet truck ZIL-I5I where it is assumed that the road between two /34
points is longer by a factor of I.5 than the air route.
Figure 1.34 shows the values of Cl as a function of flying range, for vari-
ous VTOL aircraft. The value of the flying range L in calculating Cl was deter-
mined with consideration of a decrease in fuel consunption per kilometer as the
aircraft became lighter due to depletion of the fuel. Assuming a linear de-
pendence between consunption per kilometer and the weight of the aircraft, we
must correct the value for the range, calculated from the above formulas where
the per-kilometer consunption is accepted to be constant and corresponding to
the takeoff weight, by the quantity
Ki=
log
29
I t
where _ q
G-r = — 7=r— is the ratio of fuel weight to takeoff weight of the aircraft.
For TTOL aircraft, aircraft with a large fuel consunption for hovering, the
reduction in flying weight due to the expenditure of fuel for hovering and in
horizontal flight was taken into account. Figure 1.35 shows the curve of the
correction coefficients Kl as a function of the values of Gj.
C^ kg/ ton- km
VTOL turbojet
+—— a.ircr aft
Lk
Fig. 1.34 Dependence of Cl on Flying Range L.
For aircraft with takeoff rim, the values of Gl at different fljring ranges
are given in Fig. 1. 36. The diagram also shows how the economy of transport
means can be increased, at a given
range, by using a takeoff rion. The
f(^ longer the takeoff, the larger the take-
off weight of the aircraft and hence the
greater the weight of transported cargo.
Such are the results of investigat-
ing fuel consunption for transporting
one ton-mile with various types of
transportation means.
As regards the cost of operating
airplanes and helicopters, which natural-
ly is determined not only by the cost of
fuel but also by the service life and
initial cost of the machine, we must bear
in mind that the greater power/weight /36
ratio of the VTOL aircraft conpared to
that of helicopters as well as the
presence of transmissions in some types
more or less balances this cost. As for safety in the case of engine failure
a
V
in
y
y
— -
0.1
0.1
0.i
O-hG^
Fig. 1.35 Dependence_of Coefficient
Kl on Gf .
30
diiring takeoff or landing, all advantages are here on the side of helicopters
since the propellers on an aircraft -with a pivoted vjing are not capable of auto-
rotation, and engine failiire (depletion of fuel) during landing of an aircraft
of the type Breguet-941 may lead to flow separation and an uncontrollable de-
scent because the wing is no longer washed by the propellers. Furthermore,
there will always be the difficulty of providing controllability at low speeds
in these machines.
C, kg/ ton-km
~f^VTOL turbojet
15
as
I Q-ircraft
Cros s-Coun try
Truck
wo
/YTVTOL propeller
aircraft STOL propeller
^^--J____^SS^-^ aircraft
- I — ^' '
We I icop ter
Range
2000- 3000 kxf,
I. ight transpor t
P
.Turbc
aircraft ^^fe^ _J ":''"^''°/'
. -.■ aircraft
t rack on highway
ZOO
m
m
500
Lr m
Fig. 1.36 Dependence of C^ on Takeoff Run for
Various Flying Ranges.
A further reduction in weight and fuel consunption of turboprop engines
under development at present will lead to an increase in load factor of heli-
copters. The substantial increase in service life of rotor blades, reduction
gears, and transmissions obtained in modern prototypes will equalize the amorti-
zation cost of airplanes and helicopters, after which the helicopter will become
a full and equal member of the air transportation system in its most massive
area.
6. Possibilities of Increase in Maximum Flying Speed
If the flying speed of VTOL and STOL aircraft is considered to be an int-
portant flying and tactical requirement, then the possibilities of rotocraft are
far from exhausted with respect to further increase in speed.
If we equate the power required for horizontal flight and the net power of
the engine, we can obtain the relation between maximum speed and power/weight
ratio of the aircraft:
^'„,.. = 270
N
11^,
(2.10)
31
where
M = engine power;
G = gross weight of the aircraft.
It follows from eq.(2.10) that the maxiinian speed is directly proportional
to the power/weight ratio N/G of the aircraft.
Pigure 1.37 gives the curves of the required power/weight ratio as a func-
tion of flying speed for various aircraft. The curves for heavy rotocraft show
that, to increase the flying speed above 300 - 320 km/hr, it is necessary to
sipplement the helicopter rotor with a second high-lift device - a wing; to
reach speeds above 370 km/hr also propellers are needed, which means changing
over to a rotocraft. Thus, by formulating the problem of achieving the highest
speed possible at any price, it becomes possible to decide what VTOL aircraft
config-uration to use for different majdmum flying speeds. However, it must be
remembered that the transition from heUcopter to rotocraft involves a loss in
1 T ft capacity, an increase in the cost of construction, etc. Even with a
power/weight ratio of 0.45 J^'/kgj which can presently be realized on rotocraft,
the transition from helicopter to rotocraft will not produce a gain in speed by
more than 30 - 40 km/hr.
/37
WO 200 M 1*00 500 600 W V km/hr
Fig. 1.37 Power/Weight Ratio of Aircraft as a
Function of Flying Speed.
Finally, the graph of the required power/weight ratio clearly shows the
great difference in the power/weight ratio of VTOL aircraft and of rotocraft.
At equal power/weight ratio, the rotocraft is somewhat inferior in speed to the
prcpeller-driven VTOL airplane with a short range.
32
Section 3- Basic Principles of Design
1. Selection of Engine Horsepower and Rotor Span
In most cases, the helicopter designer is siJpplied with the desired lift
capacity. Knowing the required speed, he estimates the necessary power/weight
ratio. After assigning the current percentage of the useful load ratio, he de-
termines the order of magnitude of the flying weight and hence the magnitude of
the installed power. Having selected the number of engines in view of the end
use of the helicopter (one engine for a light military machine, at least two
engines for a passenger craft, etc.), he can select the most siu-table engines
among existing or scheduled types.
Usually, it will happen that the power of the possible combinations of
engines does not match the desired power. This necessitates correcting the para-
meters of the helicopter in question, after selecting the optimton combination of
existing engines. After this, the main problem facing the designer is to select
the rotor span for the specific power plant.
How does one select disk loading?
It is known from statistics that disk loading rapidly increases with inr-
creasing flying weight and varies within 12 - 50 kg/m^ as the weight increases
from the lightest to the heaviest helicopter.
Disk loading, as a function of weight, varies even more than wing load- /38
ing of an airplane. This is of inportance since an increase in wing loading of
an airplane can be conpensated by an increase in length of takeoff run whereas,
for helicopters, the takeoff run must always remain zero.
The weight of the rotor increases approximately in proportion to the cube
of its span. However, at equal power the lift capacity of the helicopter as a
whole increases in proportion to the 2/3 power with an increase in span. In
addition, such flight data as ceiling, rate of climb, range, rate of descent in
autorotation must be inproved when the span is increased and hence disk loading
is decreased.
It is inpossible to calculate the parameters of an optimal design since
there are too many contradictory considerations that the designer must weigh.
The answer to this problem should also include a search into the past, an analy-
sis of the development of helicopters with respect to size.
How is the next (larger) helicopter to be developed?
It is obvious that the prime requisite is to increase the installed power.
However, to what extent? For exanple, if we retain the power/weight ratio (for
considerations of economy) and then, if necessary, increase the rotor span so
that the former disk loading remains, or else if we increase the power/weight
ratio and then have the opportunity to increase disk loading provided that take-
off is vertical, will we actually obtain a conparatively smaller rotor span?
The best solution is obtained with the variant having the lowest construction
33
weight, i.e., providing a higher proportion of usefiol load.
Let us find the variation in load ratio of a helicopter when its size is
increased for different power loading and disk loading.
We will examine the case in which a new larger helicopter is developed ac-
cording to the same scheme as above.
Taking the static thrust (without consideration of the ground effect) to be
equal to the weight, Joukowski's formula will furnish
r=G = (33.25;iiA^D)2/3, (3.1)
where
g = a coefficient smaller than unity characterizing the mechanical losses
of power in the transmission and those due to cooling and equaliza-
tion of the torque;
1] = rotor efficiency representing the ratio of the useful power needed
for s'upporting the aircraft in the air during hovering to the spent
power.
We will write the e:!^ression for takeoff weight of a helicopter in the form
G=Ouse +G„L + A,,7V + 0,,G, (3.2)
where
Gygg = usef-ul load of the helicopter;
G„i = weight of the nonlLfting elements of the helicopter;
G = takeoff weight of the helicopter;
ktr = an enfiirical coefficient representing the ratio of engine weight
_ and transmission to engine power;
Gjs = relative weight of lifting system (rotor with hub and automatic
pitch control mechanism) with respect to helicopter weight.
Change in load factor on increase in rotor span . We will attenpt to de- /39
n.
termine the load factor — "!," ■ of a heavier helicopter which differs from the
G
original by the rotor span:
Assuming that, in this case, the power/weight ratio N/G remains constant,
eq.(3.l), at a constant value of §T1, will yield
G3/2 = const A^D
or
34
"1/ -z^ = const.
G_ f G
nVd^ (3.4)
Hence the disk loading, i.e., the quantity
G
P= ■
JlD2
4
also should remain constant.
On increasing the rotor span, if the size of the blades changes similarly
while their number remains unchanged, the weight of the lifting system increases
in proportion to the cube of the ratio of the rotor spans:
G,,, = Gu.(|-;. (3.5)
Consequently,
Here the subscripts 1 and 2 pertain, respectively, to the original heli-
copter and to the helicopter under study (Dg > Di).
However, if the disk loading p is increased, flow separation at maximum
speed can be avoided only by increasing the loading (mainly by increasing the
n-umber of blades since a relative increase in chord is less advantageous and
causes a greater increase in weight because of the need for larger balancers to
eliminate flutter) .
In this case, when retaining the span and the tip speed of the blades, the
weight of the lifting system will increase proportionally to p, i.e.,
' PiKoJ (3.7)
or
Gi.s, = Oi,s.|^. (3.8)
Substituting this expression into eq.(3.3)» we obtain
G.S., =l-5nC-V(-f),-^^-lr- (3.9)
The coefficient k^r is the sum of the relative weights of engine and trans-
mission:
*tr=-^ + ^^- (3.10)
35
mill I II I
The first addend remains unchanged (it characterizes the weight charac- /k.0
teristic of a modern engine) and the second increases proportionally to the in-
crease in rotor span.
Actually, if we assimie that the weight of the transmission is proportional
to the magnitude of its transmitted torque M^., then we obtain
^^--^'^'m; '''''^- (3.11)
Keeping the tip speed constant (uoR = const), we have
•'I ^ D2
"2 Di
Hence,
At constant disk loading it follows from eq.(3'4) that
where
Then, considering also that
and
ffi =02/^1-
<J2 Gi
(3.12)
(3.13)
we can transform eq.(3«9) into
^"" =>-°.^-T (^-[°- +t(^)J^' (3.15)
Where ^^ ^ ,0>
— P G
Assuming for the original version: Gni = 0.25; — ^^^- = 0.2; — ^ = 0.4;
Gi = = 0.18: -^ = 0.28 and substitiifcing these into eq.(3.15), we obtain (at
G2 = Gi)
36
^G^ts. — \ =0.402;
G /I
Ous, =0.694-0.292,)/^.
(3.16)
Hence it is clear that the magnitude of the load ratio of a larger heli-
copter decreases monotonicaUy vdth an increase in m (Fig.l.38).
The ratio of the useful load of helicopters can be represented as
G,
f Guse \
o
( G use \
a A
= 2.49/ra{0.75 L^.^
-["•'«+^m]^^)-
(3.17)
It follows from eqs.(3-15) and (3-17) that it is inpossible to construct /hi
a larger helicopter while maintaining the same disk loading and the same power
to weight ratio as those of the original helicopter, with a larger (or even the
same as the previous) coefficient of the load ratio, although the absolute value
of the useftil load increases at first (Fig.l.40).
p= const
Ouse Original
hel icopter
0.3
~~-
-^
^
0.1
1
~rr
-^
0.1
-
—
n
OM
D' const
Original helicopter
Fig. 1.38 Dependence of Load Ratio on Fig. 1.39 Dependence of Load Ratio on
Scale m of Weight Increase when Scale m of Weight Increase when
p = const. D = const.
Charge in load ratio on i ncrease in power/weight ratio without change in
rotor span . Let us next examine the case where the rotor span remains unchanged
while flying weight and engine power increase, i.e., the power/weight ratio of
the helicopter and disk loading increase.
It follows from eq.(3'l) that the power/weight ratio of a helicopter should
in this case increase according to the law
aXHi).^
(3.18)
37
Equation (3.8) indicated that the relative weight of the lifting system re-
mains constant. Keeping the ratio of engine weight to engine power constant,
the relative x^eight of the engine increases in proportion to /m
"'-H-^Xi >-
m.
(3.19)
The relative weight of the transmission also changes
O.
tr
V ^ A 9.
Ym.
(3.20)
Then the expression for the load ratio coefficient will take the form
G
■A
'or
Gus-
£tng
N
N
9l
(3.21)
Substituting the numerical values, we have
G use =0.57-0.17 Vm-
The ratio of useful load to helicopter weight varies in the same manner as
in the previous case (see Fig. 1.39), i.e., decreases monotonically.
As we see (Fig.l.40), upon an increase in power at constant span, the use-
ful load increases more rapidly than upon an
increase in span at constant power/weight ratio.
Thus, it is obvious that the disk loading
depends on the weight characteristics of /42
specific engines available to the designer for
solving the formulated problem, namely to lift
a prescribed useful load. It is obvious that
the lighter the engine in conparison with a
given prototype, the greater will be the optimum
disk loading and the smaller will be the rotor
span.
This is the reason for the small loads per
square meter of the first airplanes and heli-
copters. These machines ■with their then low-
power and high-weight engines with large load-
ing were generally not airworthy.
Fig. 1.40 Ratio of Useful
Load Weights as a Function
of Flying Weight Ratio.
Thus, the designer or researcher who
■wishes to project into the future should adopt
some rules for decreasing the unit weight of
engines, rotors, and nonlifting structural
elements (by using new materials and increasing the effective design stresses)
38
aside from the possible discovery of new engine operating principles; only then
will he be able to predict the potentialities of developing larger or more
economic (load ratio, flying range) aircraft.
It woiild be natioral to e^q^ect a pronounced increase in load ratio by re-
turning to small helicopters and using the level of engine and rotor unit
weights achieved in developing the heavy helicopters of the 1960s.
Actually, a conparison of recent helicopters with turboprops of the same
weight category as the Mi-1 and Mi-4 helicopters showed that their load ratio
almost doubled.
2. Analysis of Multirotor Configurations
Sooner or later, the designer is confronted with the problem of the expedi-
ency of further increasing the rotor span and the need to change to a twin- or
multirotor configuration.
The lot production of still another blade size requires very large capital
investment for b\jilding new steel mills, presses, and other expensive equipment
required for finish-treating of spars and blade assembly. Therefore, the de-
velopment of new blades is to some extent a Federal problem. At the same time,
termination of the production of any one type of series-produced blade is im-
possible, since the existing inventory of helicopters is a steady consumer of
blades because the blade life, as a rule, is considerably shorter than the
service life of helicopters. Consequently, when initiating a new blade design
new production facilities must be created to supplement those already available.
Therefore, after having developed the largest series-produced rotor, it is
logical to attack the problem of the optimum multiple to be used. This renders
the problem of configuration specific: it becomes necessary to double or triple
also nonrotor units, i.e., rotors together with reduction gears and engines. /43
Actually, the number of combinations is not excessive: twin- rotor (side-by-
side and fore-and-aft configurations) and three- rotor helicopters. The cimiber-
some four-rotor configuration need not be discussed here since the above con-
figurations are able to provide the maximum required lift capacity of 40-50 tons.
Another problem to be discussed is that of conparing single- rotor helicopters,
designed for similar missions, with these configurations.
Fore-and-af t_ c£nf igioration . Since the induced velocities of the front and
rear rotors are identical, the induced velocity of the system will differ only
by the quantity of the average velocity of mutual induction
'"<^^=T ■'■'"" (3.22)
where
vi = induced velocity in the rotor plane;
K = coefficient of induction.
Then, the additional induced power of the system or of the rear rotor is
39
aA^,=-— y.v.G.
i 2 •
(3.23)
If both rotors lie in the same horizontal plane and do not overlap, then
= 2; at -^ = 0.2 (Fig. 1.41) we already have h = 1.35.
However, we shoiild consider that the induced velocities are unevenly dis-
tributed over the disk so that the average ve-
locities are larger, corresponding to another
aspect ratio, i.e., to a smaller span or, what
comes to the same, to a larger load p on the
sipporting surface. Therefore, we will take the
expression for the average velocity of mutual in-
duction, referred to the entire craft, as
v^^=0.75vi=0J5
4p
VA
(3.24)
Fig. 1.41 For Determining
Rotor Overlap .
Then the rate of climb of a helicopter of a
fore-and-aft configuration can be e^qjressed by
the rate of climb of a single-rotor helicopter
y
-V^
3p
(3.25)
where
Vy
Vy =
A =
rate of climb of a tandem (fore-and-aft configxjration) helicopter;
rate of climb of a single-rotor helicopter;
relative air density.
Thus, whereas the Mi-4 helicopter with blades of mixed design has a vertical
rate of climb, at a flying speed of 100 km/hr, of 3-6 m/sec, the vertical speed
of the tandem-rotor helicopter (Fig. 1.42) with two such power plants decreases
by the quantity
3-21
AVt=l/^-l/t ^'^:£i=,2.25 m/scc,
28
i.e., the rate of climb of the tandem^ rotor helicopter is 1.35 m/sec.
Consequently, the flying characteristics of the fore-and-aft machine sub-
stantially differ from those of the original single-rotor helicopter from which
the power plants were taken.
The rate of climb is determined by the transverse span of the helicopter, /l^U
its engine power, and the takeoff weight of the helicopter prescribed by the
designer. It can be stated that, if these parameters are given, the maximum
possible rate of climb will be determined regardless of the configuration of the
planned helicopter and the type of its power plant.
In addition to power e^qp ended for lifting weight, there are also power
40
expenditures for mechanical, profile, and induced drag. Thus, if the takeoff
weight of the helicopter of a fore-and-aft configuration is twice the weight of
a single-rotor helicopter, the magnitude of rate of climb can be maintained only
by increasing the power of the tandem-rotor helicopter to more than the double
power, by an amount of
^ 75K4
If we only double the power of the tandem-rotor helicopter, its rate of
climb will decrease, as indicated above. Such a doubling would be especially
unsuitable for heavy helicopters with large disk loading; thus, at values of
p = 40 kg/m^ and V = 40 m/sec the loss of rate of climb AVy at an altitude of
H = 3000 m, in conparison with the original single-rotor helicopter, will be
4 m/sec, i.e., such a helicopter will not be able to fly if Vy < 4«5 - 5.0 m/sec.
Fig. 1.42 Yak-24 Helicopter.
Cpnparative data of two Soviet helicopters Mi-4 and Yak-24 of different
configiorations but having the same engines and supporting systems (these are
doubled on the Yak-24 helicopter) show that the fore-and-aft helicopter has a
vertical groiind speed 2.6 m/sec lower than that of the single-rotor helicopter
and that, at altitudes of 1000 - 2000 m, the loss in vertical speed reaches
3-3*5 m/sec. The service ceiling also drops by a factor of 2 for the fore-
and-aft helicopter.
Figiare 1.43 shows the change in torque distribution with respect to the
rotor shafts, measured in flight on one of the fore-and-aft helicopters. At
p, = 0.1 - 0.25, the rear rotor consumes about double the power of the front
rotor.
This nonuniformity in loading of the rear and front reduction gears and
rotors substantially reduces the lifetime of the rear rotor parts or else neces-
sitates development of a more powerful and heavy reduction gear. It is inpds-
sible to use the main reduction gear of the original single-rotor helicopter as
the rear reduction gear of the tandem helicopter.
The large induced losses due to mutual interference of the rotors in fore-
and-aft helicopters, which amount to 20 - 25^ of the power at cruising speed, /45
41
greatly impair its efficiency in conparison with the single-rotor helicopter in
which the power expenditiire for driving the rear rotor in horizontal flight is
negligible .
Side-by-side configuration . This
old problem of how to build a "bridge"
/"t
_^
0.0015
\
Two ro
tors
s
V
•
•
0.0010
_--
><
Re
jr roto
-■* —
r
•
0.0005
\
'i*""^
Fron
>-
t rotoT
configuration is another solution to the
connecting the rotors . How can one de-
fine the difference between such fore-
and-aft and side-by-side "bridges"?
From the viewpoint of mutual inter-
ference during hovering, there are no
fundamental differences in the operation
of the rotors of either type of heli-
copter configuration. However, the
losses from propeller wash over the wing
in a side-by-side helicopter may be
greater than from wash over the fuselage
in a fore-and-aft helicopter. There-
fore, the characteristics of vertical
takeoff are poorer for the side-by- side
configuration with a wing. Neverthe-
less, in forward flight the rate of
climb of the side-by-side helicopter in-
creases by an amount of
Al^y = — y-iJi^ 0.3751), =
1.5p
VA
0.1
02
Fig. 1.43 Change in Torque Dis-
tribution with Respect to Rotor
Shafts as a Function of their
Operation.
in conparison with the original single-
rotor helicopter, due to a decrease of
induced drag (the "span" of the side-by-
side helicopter being twice that of the
original single-rotor machine).
It can be demonstrated that, between
the load ratio of a side-by-side helicopter and a single-rotor helicopter, the
following relation exists:
where
5g =
V G
5r e ar
=
Ne„.
5t
=
Tl
_ Gw
-
G-rear
(i-Lr[(-^^)^^-^.-m.
Tl
a coefficient taking into account power expenditures
at the rear rotor in a single-rotor helicopter taken
equal to 0.09;
a coefficient taking into account thrust losses due to
wash over the wing, taken equal to 0.07;
a coefficient taking into account the variation in
weight on changing from a single-rotor helicopter to a
two-r*otor helicopter of side-fcy-side configuration.
Let us assume that the wing weight is 12^ of the weight of the single-rotor
42
helicopter and that of the rear transmission mth rotor, 10%.
The above formula shows that, in a side-by-side helicopter constructed from
two single-rotor helicopters, it is inpossible to achieve the load ratio of the
single- rotor helicopter.
If a side-by-side helicopter is designed from scratch, it might result /U6
in a much better construction since the designer of a side-by-side helicopter
will not use the same load per square meter as the designer of a single- rotor
helicopter had been using. In fact, this factor will be made greater and the
helicopter will become more conpact. On the other hand, when designing a fore-
and-aft helicopter from scratch it is natural to select a smaller load per area
and a higher power/weight ratio than is usually done.
It must be borne in mind that, in designing a side-by-side helicopter,
there must be means available for "controlling" the frequency of natural oscil-
lations of the wing with power plants and rotors, both in the vertical and hori-
zontal plane, since this configuration has a multitude of possible vibration
modes whose frequencies may enter into resonance with the forced frequencies in-
duced by the rotor. Furthermore, in the case of high-power and thus heavy
engines moimted to the wing tips, the side-by-side helicopter almost certainly
will have a vibration mode of a frequency close to or even smaller than the
rotor rpm, at which the rotors will vibrate on horizontal displacement. This
may set tp oscillations of the "ground resonance" type not only on the ground
but also in the air. Therefore, the designer who has decided to design a side^
by-side helicopter is faced with the difficult task of making the wing as small
as possible in area, light In weight, and sufficiently rigid in bending and
torsion.
However, despite certain difficulties in designing side-by-side helicopters
and shortcomings of fore-and-aft configurations, designers will have to resort
to them as a means of increasing lift capacity. This becomes obvious when con-
sidering the difficiilties In developing rotors and reduction gears of super-
heavy single- rotor helicopters .
Selection of configuration . An analysis shows that, in changing over to
side-by-side or fore-and-aft helicopters with doubling of the power plants, it
is lirposslble to double the useful load lifted by the single- rotor helicopter.
If this could be achieved at all, it woiild be at the price of an appreciable loss
in such flying and tactical data as takeoff and landing properties, rate of
climb, dynamic celling, etc. Thus, the transition from the single- rotor to the
multirotor helicopter must be done over an increase in power/weight ratio.
However, the selection of the conflgiiration can be largely influenced by
factors such as end use of the helicopter and tactical, technical, or operational
requirements. The designer often prefers to adhere to the configuration for
which he has more data and ejjqaerience, if other conditions permit selecting
several approximately equivalent configurations. In some cases, the designer is
forced to give preference to a previously used configuration even when another
configuration might offer some advantages.
Let us give an exanple to illustrate the point. The pay load of a helicopter
43
with standard rarige amoiints to about 20^ of flying weight. Depending on the
helicopter design, this weight may vary by 5^ to either side. This means that,
at identical takeoff weight, a good helicopter will lift a payload weighing 25%
of the machine itself, whereas a poorly designed helicopter will lift only 15^
of its weight. Thus, the second helicopter will be quite inferior with respect
to lift capacity. This should be taken into account, on the one hand, in solving
the problem of selecting the helicopter configuration and, on the other hand,
in estimating the rationality of some particular configuration on the basis of
a conparative analysis of data of escLsting designs.
kh
CHAPTER II 2M
ROTOR AERODyWAMICS
Section 1. Development of Rotor Th eory and Methods of
Eroerimental Determination of it s Characteristics
The idea of using a rotor in place of a wing as a lifting system was born
in 1923' The Spanish engineer Juan de la Cierva, after the airplane of his
design had stalled and crashed, decided to develop an autorotating sustaining
system whose wing-blade combination would not lose speed at a low or even zero
forward speed of the apparatus .
An experiment carried out by him in 1924 in the laboratory of Quadro-
Ventos in Madrid, which showed the lonlikely high values of the aerodynamic effi-
ciency of an autorotating rotor as a lifting system. Induced a theoretical in-
vestigation of the aerodsmamics of autorotating rotors, carried out in England
by Glauert in I926 and later developed in 1928 by Lock for the case of hinged
blades .
Thus, experiment generates theoiy and the endeavor to extrapolate the re-
sults of theoiy to practice gives rise to new experiments which more thoroughly
reveal the pl^sical nature of various phenomena, which in turn leads to a new
close examination and development of theory. Only in the unity of theory and
practice is it possible to describe the development of rotor aerodynamics vp to
the present state of the art. Thus, in 1928 the Glauert- Lock theoiy was first
published. As is known, in this theory the magnitudes of thrust T, longitudinal
force H, and torque M^ are determined as a function of the kinematic parameter
V cos a
of the angle of rotor blade setting cp, and of the flow coefficient
V sin o — V
A:
"R
which represents the ratio of the velocity of the air flowing through the disk
to the tip speed tuR (Fig. 2.1). Consequently, the coefficients of the moments
and forces can be expressed as
(1.1)
According to the momenttmn. theorem (see Sect .3), we have /l43
45
After determining from this the induced velocity v
^ ____CrQJt/?'lo)2 _ Ct-u/?
v =
2Qn/?2U' 4QIt/?3(0 y^|i.2+ X2 4 )^(i.2+ X2
(1.3)
and substituting this expression into the formula for \,
), = jJ, tan a -
4 lA^2+ X2 '
we obtain the e^qaression for the angle of attack of the craft:
tana =
4(1 y |x-'+:):2
(1.4)
For calculating the aerodynamic characteristics of an autogiro. Lock pro-
posed deriving the unknown value of the quantity \ from the conditions of auto-
rotation, i.e., equating the ex-
pression for torque on the rotor
shaft to zero:
V cos of: „., ./,
4ufogi''°
Fig. 2.1 Velocity Conponents of Air
Flowing through the Rotor Disk.
yW^=/((p,),(x) = 0.
(1.5)
Since the angle of blade set-
ting cp and the characteristic of
the regime |j. were prescribed
quantities, the value of \ was
determined from the quadratic
equation (2.47) representing
eq.(1.5) in a developed form. For
a given value of the velocity of
rotation uuR, flying speed V,
angle of setting cp, and known \,
the only possible angle of attack
of the autogiro a at which steady
autorotation occurred was found
from eq.(1.4). In this case, the
equation (see Sect. 2)
^ = ^=/('f, !J-,/v) = a„
[\H
3 ' 2
(1.6)
yielded the value of the thrust coefficient while the corresponding equation
furnished the value of the coefficient of longitudinal force so that the polar /49
of a freely autorotating rotor could be detemdned as a function of the angle of
attack. This polar, Hke the polar of a wing, could be used in the aerodynamic
46
calculation of the autogiro. Such a method was published in 1931 by I.P.Bratu-
khin (Ref.ll).
In 1934, the author"" (Ref .4), in his study of overspeed of the autogiro
rotor during the takeoff run, determined the unknown values of thrust and torque
from eqs.(l.4), (1«5)» and (1.6) and prescribed values for the angle of attack ot ,
angle of blade setting 9, and initial peripheral velocity and flying speed,
i.e., the parameter |j,, which made it possible to calculate the aerodynamic char-
acteristics for cases of unsteady motion. The following method of determining
the flow coefficient \ was proposed:
Substituting expression (1.6) into eq.(l.4) will yield a polynomial of the
fourth power in both X and |j,; thus, it is difficult in practical work to deter-
mine \ at known ij, or vice versa, since each time it would be necessary to solve
a fourth-degree equation. A study of the dependence of X on |j, at given values
of a and cp shows that, with the exception of a small segment of negative X near
|j, = 0, the curve \ = f(p.) represents a straight line with a high degree of accu-
racy. The equation of the family of these straight lines at different values
of Q- and cp has the form
n~"-- (1-7)
Equation (1.7) yieHs the dependence ofX oniJ,,Qf,9, and a; its use, to-
gether mth the above formulas, permits calculating the thrust and torque coef-
ficients. It is thus possible to calculate the aerodynamic characteristics of
the autogiro rotor for any unsteady operating conditions when the torque created
on the rotor by the air flow produces spinning or braking of the blades, depend-
ing on the angle of attack and the value of |j, .
This still left the necessity of finding a method of applying this theory
(developed by Glauert and Lock for an autogiro) to calculation of the aerody-
namic characteristics of a helicopter, i.e., to their determination under forced
rotation of the rotor by an engine. Such a method was proposed by the author
in 1945, in collaboration with V.N.Yaroshenko (Ref .9).
To determine the torque necessary for fUght under given conditions it is
logical to use the above system of equations. Here, it was convenient to pre-
scribe the value of the thrust coefficient t, since the rotor thrust is easily
determined if it is approximately assimied that the thrust, under steady hori-
zontal flying conditions, is equal to the helicopter weight.
Using prescribed values of the rpm and determining t from the expression
^= o ^ • (1-8)
2
^ Here and elsewhere in Section 1, the author is M.L.Mil' .
47
the quaxLtity X can be found from eq.(l.6), the torque from eq.(2.47) and the
angle of attack of the helicopter a which corresponds to these conditions, from
eq.(1.4)»
Even before the present stage of helicopter development, practical auto-
giro engineering required the solution of certain stability problems; designers
in various coimtries attacked the problem of determining the damping produced
by the hinged rotor d\3ring vibrations of the craft. Analyses of flight accidents
■with autogiros showed the necessity for studying the flapping motion of blades /50
and for finding methods of retaining autorotation during aerobatic maneuvers of
the craft . This led to the theory of a rotor for hinged blade attachment with
curvilinear motion, which the author developed in 1939 and which represents a
more general case than the Glauert-Lock theory established for steady recti-
linear motion.
Finally, in 1940 A .N JVIikhaylov worked out a method of equivalent rotors,
which sinplifies the application of the Lock theory to a rotor equipped with an
automatic pitch control mechanism (Ref.l5).
The application of these methods to the aerodynamic calculation of Soviet
autogiros between 1931 and 1940 and of the Mi-1 and Mi-4 helicopters in 1947-1952
showed highly satisfactory agreement between design characteristics and flight-
test characteristics . Since these first autogiros and helicopters flew at rela-
tively low speeds and thus at small values of |j,, the inaccuracies of the theory
due to the assunption of smallness of this parameter, made by Glauert and Lock,
were nonessential. However, more coiiplex problems were still to come: the de-
velopment of more powerful and faster helicopters, which inplied constant im^-
provement of the theory.
The Glauert-Iock theory, as is known, makes a number of assunptions (in-
cluding uniform distribution of induced velocities) so as to permit integrating
the equations in a finite form. Thus, the Cy of the section was expressed as a
linear fionction of the angle of attack Cy = a^^ a, while Cxp was taken as some
average quantity independent of the angle of attack. The forces acting on the
profile, i.e., on the section of the disk where - in forward flight - the air
flows around the blade from the trailing edge (this region is small at small
values of |j.) were inaccurately determined. The radial conponent of the resultant
velocity in the blade direction was also disregarded. The blade itself was
assumed to be rectilinear, flat (not twisted), and of constant chord.
During the period of 1932-1943, many researchers - Whittle and Bailey in
the USA, Hohenemzer and Zissing in Germany, and others - further refined this
theory in that methods were foimd for integrating equations in a finite form
while doing away with many previously accepted assutiptions . The concept of the
effective radius of a blade, smaller than the actual, was introduced for taking
account of tip losses. The coefficients Cy and c^ represented more conplex
functions of the angle of attack, etc.
The most inportant inprovement of the classical theory during the postwar
years was the application of methods of n^umerical integration to the calculation
of flapping motion and aerodynamic forces. This permitted the direct use of the
e^qjerimental characteristics of profiles, taken for the necessary value of the
48
Reynolds nimbers and Mach mjmber, in deterniining Cy and Cx of the section as a
function of the angle of attack and thus to take into account also the effect
of conpressibility.
Later, it became possible to introduce into the calculations not only the
initial geometric shape of the blade but also its deformations due to bending
in the plane of thrust and in plane of rotation and, what is especially im-
portant, due to torsion.
However, the use of such a cumbersome method for practical calculations
became possible only after appearance of electronic conjjuters. Thus, the modern
method of rotor calculation, as presented in this book on the basis of studies
made by A.S.Braveiman and M.N.Tishchenko, was a step-by-step process.
However, even after all these refinements there still remained the /51
rather rough assunption of a uni-
Section where measurements
made
1
Fig. 2. 2 Induced Velocity Distribution
in Hovering.
form distribution of induced ve-
locities over the rotor disk,
which not only led to an inaccu-
racy in determining induced power
losses but also to errors in de-
termining the true angles of
attack of individual blade sec-
tions and hence to errors in the
profile power, thrust, and longi-
tudinal force.
Thus, further refinement of
the theory could be expected from
the development of the vortex
theory which is the only one
capable of determining the distri-
bution of induced velocities in
relation to the forces acting on
each given blade element.
However, development of such
a theory required greater insight
into the plgrsical aspects of the
phenomenon. Here again it was
necessary to resort to e^qjeriment.
Experimental studies of flow
around a rotor in a wind tunnel
followed by removal of the induced velocity field, carried out in 1946 \s^ the
author together with M.K.Speranskiy, clearily showed that the vortex system known
from a propeller operating under conditions of axial circulation flow and repre-
senting [for the case of circulation constant along the blade (F = const)] a
central vortex with blade-tip vortices shed by the blades, is transformed - at
small values of p, - into a system similar to the rectangula.r vortex system
characteristic for a wing. In turn, the induced velocity distribution obtadLned
from experiment (Pigs. 2. 2, 2.3> and 2.4) fully confirmed the possibility of an
49
0.1S
Tr
"P
ezoida
bladc
A-
az u=-5'
a:
i°^
rn/sec R
f
\0.S
-I
s
as
V
^=0
as
-1
-2
0.5
I
Z^
Fig. 2. 3 Induced Velocity Distribution in
Forward Flight (|j, = 0.2).
approximate representation of the induced velocity field, proposed by Kusner and
Glauert, in the form of a funnel during the hovering phase (Fig. 2. 5) and in the
form of a cylinder trimcated iDy an inclined plane during forward flight
(Fig. 2. 6). This configuration could be used for refining the Glauert-Iock
theory Td.thout resorting to the vortex theory. These e^qDerimental facts gave
rise to a sequence of theoretical works, which were based both on the approxi-
mate vortex pattern of rectangular vortices suitable for describing conditions
at medium and high flying speeds
and on various more general
theories which examine a system
of vortices trailing from each
element of the blade.
It is necessary to say that
even before appearance of these
ejgjerimental data, G.I.Maykopar
examined the vortex theory of
rotors, having proposed that the
vortex cylinder slopes in direc-
tion of flight, which then
served as incentive for a series
of more or less accurate studies
in this area.
Fig. 2. 4 Three-Dimensional Model of Induced
Velocity Field of a Rotor.
Subsequently, L.S.Vil'dgrube
was successful in developing the
vortex theory in the USSR, fol-
lowed later by V.E.Baskin.
50
In all these theories for determining the angle of attack of the TDlade /53
section it was necessary to define not only the known kinematic parameters o?, p.,
i|r, and 9 but also the vertical coirponent of the induced velocity v as a result of
the action at a given point of all vortices in the region surrounding the rotor.
The induced velocities are found isy the Biot-Savart formula as a function of the
circidation in the blade section:
Hence it is clear that, since the induced velocity determining the angle of
attack of the section is a f\inction of circulation and since the latter, in
turn, can be determined in terms of Cy which is a function of the angle of attack
of the section in which the induced velocity enters, all these problems reduce
to quite conplex ecfuations and can be solved only by approximation methods and,
in particular, by methods of successive approximations . Generally, first the
magnitude of Cy is determined under the assimption of constancy in the distribu-
tion of induced velocities over the disk, after which the calculation is re-
peated during which process the induced velocity is determined from the vortex
theory. The integral value of the coefficient of thrust is then conpared with
the assigned value, and leads to calculation of successive approximations.
Vr'Kr-
V COS ct
Plane of rotation
^(^,f) = v+arcasfj
Fig. 2. 5 Induced Velocity Distribution Fig. 2. 6 linear Induced Velocity
according to the Law of the Triangle Distribution.
Without here examining the essence of various vortex theories which, spe-
cifically, differ by the assunption of finiteness or infiniteness of the number
of blades, or the principles of methods for calculating the induced velocity
(see Chapt.II, Sect.5), we will estimate their general significance and role in
the refinement of calctilations of the flying characteristics of the single-rotor
helicopter.
Let us examine the maximum value of the air in determining the induced
power, assuming a unifonn and funnel-shaped induced velocity distribution.
After this, we will conpare the induced rotor power under the condition of con-
stancy of thrust for the case of a uniform induced velocity distribution over
the disk and corrections for the funnel.
The funnel-shaped induced velocity distribution is represented by the law
51
of the triangle (Fig. 2. 5):
where v^ is the induced velocity at a radius r of the blade.
According to the momentum theorem, the elementary thrust is /54
dT=Q2nr2v'i.
Integrating from to R, we obtain
where v^ is the induced velocity in the blcLde tip section.
For the case of uniform induced velocity distribution, the thrust is
7",=q2jt/?2z,2_
Equating Ti and T, we obtain
Qn/?2t)2=2jr/?2Qx;2,
whence
v^ = vy2.
Consequently,
v, = v^ —=.v— y'2.
The induced power is determined from the following formulas:
dN-^„^ =dTv = AnQv\rdr\
R
Substituting into the last formula the value of Vj. = kr, we obtain the ex-
pression for power in the case of an induced velocity distribution according to
the law of the triangle:
A^i„4 =— /?2Q23/2'r;3=2.26n/?2Qx;3.
tri 5
This power is greater by a factor of 1.13 than the power Nj^^ for the case
of -uniform velocity distribution, since
52
Thus, the funnel-shaped induced velocity distribution increases the induced
losses by about 13%' Consequently, the difference in the relative efficiency
amounts to about 10^. This is a high value so that, for calculating the thrust
and power of a rotor in a hovering regime, it is logical to use methods of the
momentiom or vortex theoiy which permit talcing into account the additional power
loss due to nonuniform induced velocity distribution over the disk.
In an approximate calculation, the induced losses calculated by the formula
can be increased by about 13%.
In hovering, the expended power is the sum of induced power and profile
power, where the former amounts to 75^ of the total required power. At cruising
speed, the induced power is only 20 - 30% and at maximum speed, about 10^. Thus,
at high flying speeds the maximim refinement in the required power, as a result
of taking the nonuniform induced velocity distribution into consideration, /55
can be not more than 1 - 2^ in power and not more than 1% in flying speed.
Of course, using (analogous to the effective aspect ratio of an airplane)
an effective rotor radius somewhat smaller than the actual radius, it becomes
possible to introduce an experimental correction to the aerodynamic calculation
which had been based on the theory of constant induced velocity distribution.
This correction may be particularly inportant in determining the rate of climb
and ceiling for heavily loaded rotors.
Thus, it is obvious that, in calculating the flight data, there is no need
for much more conplex calculations of the aerodynamic rotor characteristics
based on the vortex theory.
Refinement of the section angles of attack given by the vortex theoiy be-
comes necessary only in calculating the stresses set vp in the blade, especially
at low speeds where the aerodynamic forces inducing blade vibrations in second
and higher harmonics actually are the result of the blade encoixntering the
vortex field. Accoixiing to the theories stipulating v = const, these stresses
- for all practical purposes - are equal to zero.
It must be assimied that further refinement of the velocity field may be of
importance in determining the boundary conditions at which flow separation be-
gins. However, for this purpose it suffices to refine only the pattern of the
induced velocity distribution in the form of longitudinal pitch, which leads to
a variation in the flapping motion (bi) and to some redistribution of the angles
of attack.
At the same time, the concept of induced velocity distribution in the form
of a funnel (Fig. 2. 5) and especially the assunption of its b-uildiqp from front
to rear (Fig. 2. 6) result in substantial variations in the flapping motion (spe-
cifically in bi ) and in the lateral force S, which has been taken into accoiont
by various authors.
53
1. Classification of Rotor The ories
It is obvious from this account that only two of the theories of a real
rotor differ fundamentally. Both are based on a study of forces acting on the
blade element. Such an approach to the helicopter rotor was first used by
N.Ye.Zhukovskiy (Ref .1) as long ago as in his investigations on the effect of
wind on the thrust of a helicopter rotor.
In the first of these theories, the induced velocity distribution over the
disk is prescribed, regardless of the forces acting on the blade elements. Its
average value can be determined from the moment lan theorem.
In the second theory, the induced velocities of each blade element are a
function of forces acting on all blades, which in turn are a function of these
induced velocities and are usually determined by means of the Biot-Savart
formula.
Let us call the first the classical theory which enconpasses the Glauert-
Lock theory and its subsequent development, while the second will be designated
as the vortex theory.
The classical theory is conceivable with integration in finite form, while
the vortex theory proposes only numerical solution methods.
It is also of use to study the momentum theory of an ideal rotor, which can
be used in developing energy methods for aerodynamic calculation and in inter-
preting the results of an experimental determination of the aerodynamic rotor
characteristics .
Although the development of the classical theoiy in its numerical methods
is nearing conpletion due to the use of conputers, the vortex theory still /56
presents various problems. Thus, in most vortex theories, when calculating the
circulation flow, the quantity Cy is taken as a linear function of the angle of
attack since it can be refined. For sicplification, the vortex system is con-
sidered to be two- or three-dimensional but linear. This sinplification can be
conpensated later.
Thus, the vortex theory, having inherited all refinements introduced dioring
the development of the numerical calculation methods of the classical theory, is
presently becoming the most accurate theory. In its development, there is no
need to use the assunption of steady flow around the blade sections and the
section polar can be refined Isy using ejgjerimental data as to the influence of
centrifugal forces on phenomena in the boundary layer.
Proceeding from this classification, let us give a further account of the
rotor theory.
2. Development of Experimental Methods
Ibiperimental methods for determining the aerodynamic rotor characteristics
are being developed simultaneously with the described development trends of the
54
r
theoiy.
After the first experiments in Madrid in the 1930s, researchers in mar^r
coiintries began experimenting vdth rotors in wind tunnels. Experiments based
on procedures by V.G.Petrunin at TsAGI were carried out (1931-1936) on models
of autorotating rotors of 1.2 m diameter, in which the three conponents of force
and torque were measured and the blade flapping recorded.
The experimental technique was greatly inproved. In particular, it became
possible to obtain the fuselage polar in the presence of an operating rotor.
Measurement of flapping at the hub of an autorotating rotor, tiurning in the
tunnel air stream about the translational velocity vector [which permitted esti-
mating the rotor danping in roll (see Sect. 2)] was the most noteworthy achieve-
ment by V.G.Petrunin in these e35)eriments .
However, it soon became clear that small Reynolds numbers in the case of
flow around the blade section led to such extensive distortions of the profile
polar on the model, in conparison with a full-scale craft, that it was impos-
sible to use these results directly.
In 1944, the author together with I.F.Morozov at TsAGI set up experiments
on a rotor of D = 2.5 m, which no longer concerned only the autorotation regime
but also included the helicopter regimes .
The testing facility for the rotor model of 2-5 m diameter is shown in
Fig. 2. 7. Tests on this model were first made in the coordinates 6, mt = f(cp)
for Li = const and t = const (see Fig. 2. 8), where 6 « ,^ . The tests, conducted
for three types of blades of different shape and twist, made it possible to
judge the propulsive properties of the rotors. However, the main thought behind
these experiments lay in the possibility of estimating the degree of perfection
of the theory in conparison with calculations.
Fig. 2. 7 Testing Facility for Rotor Models.
55
I mil II iiiiiii 11 1 mil 1 1
W.
n f
-^
p^
~
[ —
p-
.0
^:
^?
-
—
-
—
—
—
—
A
"^
•-"
i2,!2f
S./ff
0.15
DlOcp
Whereas our first helicopters were calculated with reasonable acciaracy by
the classical theory presented in Section 2, the Mi-6 helicopter vrLth lljOOO-hp
turboprop engines had already passed into a region of flying speeds - and hence
of values of [j. (more than 0.4) and Mach numbers - where the Glauert-Lock theory
yielded appreciable errors. In preparation of this, a full-scale helicopter
test stand was constructed at the TsAGI on which the author, together with M.K.
Speranskiy, determined the characteristics of l4.5-m rotors in a large wind /57
timnel. The next problem was that of
converting the e^qDerimental data ob-
tained for a specific rotor to another
load factor and to other Mach numbers.
These calculations are presented in
Section 6.
Because of the need to refine the
theoretical methods of calculating
aerodynamic loads on a blade element,
a series of experiments were carried
out in recent years, whose results
jrield a more detailed explanation of
the circulation flow pattern around
the blade profile under actual rotor
operating conditions. In this respect,
interesting data can be obtained by
measuring the pressure distribution
over the blade chord in flight. By /5S
integrating the pertaining pressure
distribution diagrams, it is possible
to obtain the actual inr-flight aero-
dynamic loads acting on the profile
and varying relative to the rotor azi-
muth. In making such experiments on
models, V.E.Baskin and A. S.D Dyachenko
found that, as the blade passes into the region of vortex trailing from the tip
of the advancing blade, there is a marked junp in blade loading.
No doubt, the further development in helicopter engineering toward an in^
crease of fl3ring speeds will require new experiments and the development of test
stands for experimental determination of the aerodynamic characteristics at even
larger values of (j, while retaining similarity of Mach and Reynolds numbers.
Section 2. Classical Theory of a Ro tor with Hinged Blade
Attachment; General Cci.a eL Curvilinear Motion
The rotor theory for rectilinear motion of a helicopter or autogiro was de-
veloped by Glauert and Lock and has been described at numerous occasions [see
(Ref.36, 37, 2, 11, 23)]. The theory presented below is a further development
of the Glauert-Lock theory, for the more general case where the helicopter is in
ciirvilinear motion so that the rotor axis describes a rotary motion in space.
Such motion takes place during steady ciirvilinear flight of the craft, for ex-
anple, during turning and also during vibrations relative to the longitudinal or
-o-t-0.1; -a- t=- 0.125 -Trapezoidal blade
— t-O.I; •^■t - 0.125 -Rectangular blade
Fig. 2. 8 Aerodynamic Rotor Charac-
teristics (Model Test).
56
transverse axis caused isy piloting or external factors.
Rotor Theory in Curvilinear Motion
Here, all the assxmptions made by Lock are accepted in the general theoiy,
namely: The induced velocity in the absence of rotation of the rotor axis is
considered as uniformly distributed over the rotor disk; the change in the law
of its distribution over the disk pertains only to the siperposed effect of ro-
tation of the entire craft, i.e., the initial pattern remains the same as for
lock^s pattern.
likewise, we adopt the assun^jtions of linearity Cy = f(ff) and the admissi-
bility of replacing the coefficient Cx by some average quantity c^p which is
identical for all blade sections.
We will take into account blade tip losses, i.e., consider that no lift is
developed at some tip portion of the blade and that the drag forces, just as the
inertia forces, act on the entire radius. The method of obtaining the expres-
sions for forces and moments here is analogous to the Lock method, so that we
will not repeat the derivation of the fundamental equations here and only dis-
cuss expressions that differ in the case of curvilinear motion.
A special role is played here by the Coriolis forces of inertia arising
upon rotation of the rotor. These forces will be discussed in most detail.
The resiolts to be presented here contain the basic relations of the
Glauert-Lock theory for rectilinear motion and, furthermore, permit answering
the following questions:
a) What is the flapping motion of the blades in curvilinear motion of
the entire craft?
b) How does the position of the aerodynamic resultant change in the case
of rotation of the entire craft in some direction (for exanple, to
the left or to the right) and is there a tendency to accelerate or
decelerate this rotation?
c) What peculiarities does the autorotation regime of a rotor exhibit /59
in the presence of rotation?
Along with an investigation of these problems we will also discuss the ef-
fect of the configuration of the hub, profile, and blade centering on the be-
havior of the rotor from the viewpoint of stability and safety of the craft.
The obtained resiilts are common for any rotor with hinged blade attachment,
be it the rotor of an autogiro or of a helicopter.
In addition to answering the above questions which are of independent in-
terest, the res\J.ts of the analysis yield some necessary data for studj^ing the
controllability and dynamic stability of the mentioned craft.
1. Coordinate System and Physical Scheme of the Phenomenon
Coordinate system . The phenomenon is examined in a coordinate system fixed
57
vd-th respect to the craft (Fig.2.9)« This is a right-handed system (for right-
hand rotation of the rotor) in which the z-axis is directed along the axis of
rotation of the rotor and the x-axis backward with respect to the direction of
the velocity of the center of gravity of the craft. The angular position of the
blade ;jr is reckoned from the x-axis.
Physical scheme of the phenomenon . We will study a rotor rotating in space
together with the craft at a constant angular velocity and, in so doing, main^
taining a constant angle of attack with the fUght path.
Such rotation occurs, for exanple, in turning.
Fig. 2. 9 Coordinate ^stem.
Fig. 2. 10 Velocity Diagram of Rotor
in Turning.
We are interested here in the manner in which the radius of rotation of the
craft is to be taken into consideration.
For this, let us examine a helicopter executing a turn of radius p at a
large angle of bank Yt > close to 90° .
The velocity diagram in the plane of symmetry of the craft, in this case
close to the turning plane, is shown in Fig. 2. 10.
The relative flow velocity at a distance r from the center of the rotor can
be divided into two conponents: velocity Qp directed along the tangent to the
path of the center of the rotor hub (we will call this the tiu"ning speed) and
velocity ^r directed perpendictilar to the rotor plane.
The distance from the center of the rotor hub to the center of gravity of
the helicopter will be neglected for the radius of ctirvature of the path.
If we now transfer the center of rotation of the craft to the center of /60
the rotor hub, it should become possible, in addition to the indicated veloci-
ties, to also allow for the additional centrifugal forces acting on the blades.
58
As is known, in the case of a steady turn without sideslip, the resultant
of the centrifugal force and the force of gravity lies in the plane of syiimetiy
of the craft.
Thus, these forces, proportional to C?p, can be regarded as an increase in
blade weight. Obviously, this increase is the overload n.
The expression for the coning angle ao contains a term taking the blade
weight into account:
(2.1)
where S^.i, is the static moment of the blade weight relative to the horizontal
hinge.
For turning, we thus have
^%- /,-.V (2.2)
It is known that, in turning, the revolutions of the rotor increase. If
(Dt = ('u/rT, which roughly takes place during autorotation, then, substituting cut
into eq. (2.2), we obtain
Atto^Aao^, (2.3)
i.e., dioring steady curvilinear motion of the rotor without sideslip, the de-
crease in coning angle due to the weight of the blades remains constant regard-
less of the overload.
In the usual case, Aao is not more than 0.3°; this quantity can be either
neglected or taken into accoimt, regardless of how this is done in the theory
for rectilinear motion. Therefore, we will here discuss the following scheme:
The rotor moves at a constant angle of attack and executes a rotary motion rela-
tive to the axis going through its center.
2 . Ine rtia Forces Acting on the Blade
At curvilinear motion of the helicopter, the rotor blade executes four
rotary motions in space.
First, it rotates about the hub axis Oz with an angular velocity uu ; second-
ly, it rotates together with the axis Oz in space with a velocity Q having the
conponents 0^ ^^d Oy; finally, it vibrates relative to the axes of the flapping
and drag hinges making an angle p = f (f ) with the plane perpendicular to the hub
axis and an angle § = f(ilr) relative to its own mean position.
Below, we will neglect the blade motion for the drag hinge, in view of its
relative smallness .
Let us examine the elementary inertia forces acting on the blade diiring
59
rotation of the rotor. The centrifugal force C
dC=ma^rdr (2-4)
is perpendicular to the rotor axis and directed along the radius .
The centrifugal forces generated during rotation with angular velocities /6l
Qx and Qy vd.ll be
dCs^ = mQ'^^rs\nifdr, i
dCsy=mQlrQos'^dr. \ (2.5)
These forces also lie in the plane of rotation and, accordingly, are per-
pendicular to the vectors Qx and Qy
The inertia force of flapping, perpendicular to the blade, is
dj^m^-^rdr. (2.6)
The Coriolis inertia forces K produced by the rotations cu, Q^, and Qy, per-
pendicular to the plane of rotation, are
dK^^-= — 2ffi2^u)r cos <jj dr\ \
dK^ = — 2/re9y«)r sin <jj c?r. J (2.7)
The Coriolis inertia forces generated during blade flapping, i.e., during
rotation with an angiiLar velocity Q, are as follows:
- conponent in the plane of rotation
-^r dr\
(2.8)
^ "^ " dt
dKls=-2mQy^rdr;
y ^ dt
— conponent perpendicular to the plane of rotation
(2.9)
^ dt
dKls =~2mQy<^-^rcos'!fdr.
The Coriolis inertia forces due to flapping and rotation at angular velocity
at/Cp„= -2/reo)B ^rdr (2.10)
lie in the plane of rotation and are perpendicular to the blade.
60
The diagram of the inertia forces generated during curvilinear motion is
given in Fig. 2. 11. The forces dC, dJ, dK'^Q
and dKgQ are not shown.
In studying the blade flapping relative to the flapping hinge, we will take
into account only forces with respect
to eqs.(2.4), (2.6), and (2.7), since
any forces dKgQ [eq.(2.9)] of a higher
order of smallness can be neglected
for the inertia forces of flapping or
for the forces dKQ and since also all
centrifugal forces produced by rota-
tions Ojc and Qy [eq.(2.5)] can be
neglected for the principal centri-
fugal force [eq. (2.4)]; the ratio
Fig. 2. 11 Diagram of inertia Forces.
-jjj-1 , for all practical purposes, is
not greater than 0.01.
These centrifugal forces /62
Ceq.(2.5)] are accounted for in
studies of the equilibrium relative to
the axis of rotation in the e^qjression
for torque.
In addition to inertia forces, aerodynamic forces act on the blade element ,
With the aim of obtaining
a clearer view over the effect of inertia forces
and to check whether the general equations de-
rived below correctly describe the phenomenon,
we will first siitplify the problem.
We will coii53letely disregard all aerody-
namic effects and study the motion of blades
hinged to the hub and rotating by inertia in a
vacuum while their axis of rotation is turning.
Motion of hinged rods rotating by inertia
during hub rotation. Let us examine the general
case of a hub whose horizontal (flapping) hinges
are at
tion.
some distance t^
from the axis of rota-
Fig. 2. 12 For Determining
the Flapping Motion of the
Rotor .
Let the rotor rotate about the z-axis at an
angular velocity ou (Fig. 2. 12).
The axis of rotation z turns backward at a
constant angular velocity Q, and, at some instant of time, is deflected through
the angle 5 from the original position Zq. Taking, as reference plane, the plane
perpendicular to the z-axis and assuming smallness of the angle 6, eqs.(2.4),
(2.6), and (2.7) can be used for deriving the expressions for the moments of the
acting forces relative to the axis of the flapping hinge.
61
The moment oi" the inertia forces of flapping (with the positive moment
tending to raise the blade ijpward) reads
^j=- ^m^ir-l^,J^dr =
'kk
f R R R
"WW
'h.h 'k.W
Since, for modern helicopters, the ratio — ^'J' ■ usually is not more than /63
0.02, we can assume that
^ mr^dr^^mr-^dr^I^^; (2.11)
R R
j mrdr^^mrdr^S^^, (2.12)
'w.y,
and since the integral containing t^^^ can be disregarded, we have
Mj--^{I^.^~2S^J^,^.
Having designated
u,, (2.13)
we obtain
'WW
'w.w
*'=-'- S-('-2.). (2.^)
The moment of centrifi:igal forces is
R
M,
(.= -Jp/ra<oV(r-/^J^r=-/v,.,«)2p(l-E). (2.15)
Here, we assume the angle p to be small and consider that the vector of
angular velocity w is directed along the axis of rotation z .
The moment of the Coriolis inertia forces is
R
jM
K9=-2jOTQyO)sln<J;r(r-/i,^,)fltr=-2/j,j,Qy«)(l-E)slni}>. (2.16)
Equating the sum of all moments to zero, we obtain the equation of blade
motion:
^WW ^(1 -2^)+^v,.r=?(l -^)= -2/w.W V(l -^)sin'l'-
62
After dividing hj (1 - eT t neglecting e^ , and setting u) = -gl-, we finally
obtain
^-|-P(l + s)=-2^(l + s)sini.. (2.17)
Let us find the particular solution of this equation. Setting p = N sin i)f
and substituting into the equation, we obtain
2Sv
p= ^(l+e)sin.l).
The general solution of this equation will be
22
p=/lcosl/l + e<i.+ Bsinl/l+e'l' (l+e)sin<j;. (2.18)
Substituting the initial values of i|f = -^; P = 0; -^— = 0, we then /6k
determine the values of A and B:
2Qy , Jt
£(0 2
22y , It
fi = ^^(l+s)sinVl-(-c— .
(2.19)
Substituting these values into eq.(2.18) and assuming ti = t ^ (here,
■ill is the angle reckoned from the position of the blade at the initial moment,
when t = O), we obtain
f!^?^(cosl/r+lti-cos4',)(l+0- (2.20)
£0)
From this, it is easy to obtain also the solution of p for the case of
hinges intersecting the rotor axis (l^.n - ^ = O) .
We will evaluate the indeterminate form by the L* Hospital rule:
22y — (cos/TT^h-cos+i) 2^
%^o= lim -•= 4'iSin'Jj,.
Since the rotation is assumed to be uniform, then, having introduced i|fi =
= cut, we find
8 = 2< = ^^i, (2.21)
and, substituting \|ri = i|f 2_, we obtain finally
63
P = 'ij COS i]) = 8 COS i|j .
(2.22)
This result can be obtained also by direct solution of the differential
equation (2.17) in which it is assumed that e = 0.
It follows from eq.(2.22) that, in the case of a hub with intersecting
hinges with the rotor axis deflected backward at an angle 6, the blade will be
deflected in the opposite direction through the same angle.
This means that the plane of the blade tips in space remains constant while
the rotor hub turns.
The result is correct and has physical meaning.
The solution for p is plotted in Fig. 2. 13 in the case of siperposed (e = O)
and spaced hinges (for e = 0.2"; at -yj^ = 0.01. The curves were constructed
on the basis of eqs.(2.20) and (2.22).
/?i 1 1 1 1 1 [ 1 1 1 PT ' ~ • rrr "1 m ' '"T ur
^^L " ' -^-^-ZiZ
— " " ' 4 '1
^^■f^f r ' /^ '
n -*"'""' l—\ <-» -■^ 1
_-- — *^ '~ \ ] 7^1
IT -'^^ f7i-\ 7^1 '~ /JV l: / \\ ' .:
LL _^--?^. n\\ tlX y. ^ l\\ J \\ \
^-«J 1L\^^^. l^\' ~t ' -X - /. - 1 -, 1 \ l-
n ^-^-"^ % 5 ^ ' ^^ r X. j^L _i i:^ .\.
'' ■ 9r JM'm ist'isi JkUBJO 720 mi ^oo m m. '^ im '^wo^ moisjo mo mo im mrzoio ip .
-^^^t/^ A 71 \Z, T i' \-^L \ L_L I.
— -^--^ \ r i^r; \r/ \i ^^\
^-^-^^7 u, I:: \t;'l i^^t i
-f? " -^ ^< ' - \i ^ ^v^ ^
— " ' - \\"^~--X^ 1 :5 ' v/ / J
■ —— — Hinges auperposed -. . _^ ^^ i y
■**■
^^ I J_ 'aL
Ihl
Fig. 2. 13 Flapping Angle p vs. Azimuthal Position of Blade.
As indicated by Fig. 2. 13) over a certain interval of time the motion of the
blades almost coincides, i.e., their position in space is constant; after this,
the blade on a hub with spaced hinges gradually starts following the axis of /66
rotation, i.e., the plane of rotation of the blade approaches the plane of rota-
tion of the hub, with some phase shift.
^ Such a large value of e is used to make the graph clearer. In practice, e
does not exceed 0.02.
64
At this time, however, the angle of turn of the hub is already so great
that a further analysis is useless since" the blades are pl:ysically restrained in
their arresting devices on the hub and since the above theory holds only for
small 6 .
Thus, the study of short-time oscillations shows that the rotor with small
hinge spacing is practically equivalent to a rotor with intersecting hinges,
i.e., it retains its position in space unchanged.
S?r
sn
Fig. 2. 14 Velocity Diagram of
Blade Element .
Fig. 2. 15 Change of Velocity Field
during Rotor Rotation.
In the subsequent analysis we will disregard hinge spacing, and assume
e = 0. Furthermore, we have reason to believe that the aerodynamic forces (as
will be shown below) entrain the plane of rotation of the blades beyond the
motion of the hub so that equilibrium is established between the blades and the
inertia forces that tend to keep the rotor position in space constant; thus, the
plane of rotation follows the motion of the hub with some lag.
3. Aerodynamic Forces Acting on the Blade
Let us assume that the aerodynamic forces acting on a blade element lie in
a plane perpendicular to the blade axis and depend only on the velocity com-
ponent lying in this plane.
The coordinate axes Ijning in a plane perpendicxolar to the blade axis are
situated such that the Ox axis is parallel to the plane of rotation and the Oy
axis, perpendicular to it, Hes in a plane containing the rotor axis.
Let us now deconpose the velocity acting on the blade element into the comr-
ponents U, and Uy (Fig. 2. Iff-) . These are equal to
U, = XRm — r
C/_f^r(o-i-(i/?(B sirup;
- (ji^cBp cos <^ -\- Qyf cos (J) — QjT sin <]).
(2.23)
(2.24)
Here, eq.(2.23) is the same as in rectilinear motion. In eq.(2.24) only
the last two terms are added, which take into account the effect of rotary motion
of the axis. Consideration of these terms gives the change in Velocity field of
the rotor, which becomes analogous to that depicted in Fig. 2. 15-
65
.^ Taking the new values of U^ and Uy into
dU^ account, all elementary forces acting on the
rotor blade can be obtained.
As derived by Lock, the elementary /67
aerodynamic forces and moments are expressed
in terms of velocity conponents of the ele-
Fig.2.l6 Forces Acting on a ment in the following manner:
Blade Element.
dT =-i- M- (?f/^ + t/,f/,)rfr; (2.25)
dM = ^bQ (c,^^^ Ul - a^^U^U, - a^Ul) rdr; ( 2 . 26 )
ar/y = ^^sin<j.-pflrrcos^; (2.2?)
dS== ^cos(j> — pafrsintj^. (2.28)
4. Equation of Moments ReIlative_to Flap:oing Hinge
The equation of moments relative to the flapping hinge can be written as
follows (see scheme in Fig. 2. 16):
R
— r ^ OTr2 oTr — r pOTu>V2 dr—^ 2mQyW sin ipr2 dr
R BR R
— f 2m2_fU)COs<Jjr2afr4- {dTr~ { mgrdr^O.
R BR R V,2.27J
After integration, we obtain
BR
let us change from differentiation with respect to t to differentiation
dt
with respect to i|f, assuming j|. = u) :
dl_dl_ _ d^_d2l_ 2
dt ~d']i ' dfi ~rf<j/2
and then rewrite the equation of moments, after dividing it by I^.^uu^ :
BR
rV^~~ \ ^' ' '^ sill y ^ (-UOV . f ^ r^r^\
''•1^ ^h.K"0 " - /kK"2 (2.30)
66
Using eqs.(2.23) and (2.24), we write out the e^^jressions for U^ and U^Uy
entering eq.(2.25) for dT:
Ul = rW -f 2oy^Rr^ sin tj) -j- ^^R'^ay' sin^ i);; (2.31)
i/^f/y =Xu)2^r — ^(or2 — p(i<o2/?r cos ijj — 2^u)r2 sin 4. +
-f X(x(b2/?2sjha ^ !iu)/?r sin i]j— SaWy?^ sin il) cost!) —
''^ (2.32)
— 2_jU)(i/?r sin^ i[) -(- GyUir^ cos i{» 4" 2y«>(j./?r sin <(< cos >]).
The thrust . moment is
BR BR
I B2 „ . „ , , B3 B4 rfp B3 B< Q^ , , ,
+ — cpp,2sin2,^+— X ^__p(j.cos<^-— --t sin(p +
2 3 4 rfip 3 4 o)
, B2 B2 B3 rfp . , (2.33)
-| X[j. sm <jj ■ P|j.^ sin i}) cos ijj !- (a sini|) —
B3 Qr B-l 2v . B3 Sv
3
— (J. sin2 J; -| cos <jj -| |j. sin dicos ij< ) .
<i) 4 o) 3 to /
boa R^
Let us substitute expression (2-33) into eq.(2.30) and set y = Jl "^ — ;
this finally yields ^l^.h
, B4 2y B'' 2y 1 B4 2 (2.34)
H — ^ cos(;;+~ [j,sin2cjj-| -cp^ B^fu.sin'!j +
4m 3io2 43
-| (f\>.^sm^<l^)~2 — sint}) — 2 — cos'l'
2 / M (O ^v, t, '
The particular solution of the differential equation (2.34) can be repre-
sented as the series
fi==ag — aiCOst|i — i'lSint}) — a2C0s2(j) — 62sin2i]) — . . .
It is known from solutions obtained with retention of only five terms in
the expression of p and from practical experiments that, in the usual case of
(n = 0;, the second harmonics of the angle p are small in conparison with the
first.
For greater clarity and sinplicity of the derivations (while fundamentally
retaining an acciu-ate pattern) we will discard, in solving the problem, the
second harmonics in the expression for g, i.e., we take p in the form
p=ao — aiC0S(;»-6iSin<|;. (2.35)
67
We then find the derivatives -gr- a^nd '^ g -- and substitute them into
eq.(2.34). Discarding terms containing functions of double angles, we derive
-[Mf-(5=+-f!^^)-f-«oYl^]cos.], = Y[f ^+f 5^fiH^^^)]-
SLL , /B2 B4 Q r 2 D3 o ^y\ • , I (2.36)
/^^(i)2\2 4<o o (1)/
Since eq.(2.36) is an identity, we equate the coefficients of sin ^ , /69
cos i|f, and of the free term, whence we obtain the coefficients of flapping
motion.
The free term is
„,.v[i|.+ Xsn.H,.')-«^.]-,^. (,.3,,
The coefficient of sin i|f is
B2 . 1 X B2 BiQ^ 2 Q„
from which it follows that
1 /n.^^ , 8S:
The coefficient of cos i|f is
B2-^|.2 ^ •"' B2^B2_i-^2^
B'2 / 1 \ B3
4<o
whence
4 B / Sv 82
^=Yl^«o — (^*-f- ■
S'lQyY 2 ?:L
1 \<o Vw// 1\
B2+— 1^ \ T^ /B2^62+— H.2J (2.39)
For the conditions of rectilinear motion we obtain
68
a\ =2^{\^±. 5cpj
K=^ V-°Q
B2_^.2
B2 + — ^2
Then eqs.(2.37), (2.38), and (2.39) can be rewritten as
I (0 yo) I
6, = ^;-(fi-.^-«?i
' <i) ya
52(52—1-^,2)
52(b2 + ^^2J
(2.41)
¥e have examined the particular solution of eq.(2.34). Investigations of
the general solution of this equation characterizing the natural vibrations of
the blade show that the blade is stable and, after having been deflected from
its path normal for the given regime, will return to it under strong danping
during one or two revolutions of the rotor.
This means that the new flapping motion of the blades, described by
eqs.(2.4l) and caused by the presence of angular velocity of rotation of the
craft, is established rather pronptly (within one revolution of the rotor).
/70
The obtained expressions hold for rotors with large y. Equation (2-41)
shows that, imder the effect of angiilar velocity, the cone of the blades rotates
together with the shaft, lagging behind it by a constant angle Aa^ and is also
deflected in a perpendicular direction. Thus, the plane of this new slope is
shifted in phase relative to the plane of rotation of the rotor axis.
This shift h^ in the case of longitudinal rotation is characterized by the
relation
tan Af]* = — !• ,
where Abi and Aa^ are determined by eqs.(2.4l).
Equations (2.4l) can be used for conpiling a table (see Table 2.1) showing
the direction of flapping of the cone under the effect of angular velocity.
Deflection of the cone in the case of conplex motion of the rotor (for
exanple, to the front and right) is easily obtained by means of eqs.(2.4l).
Thus, for the particular case y = 8 and Q^ = Hy we obtain Table 2.2.
69
li
TABLE 2.1
TABLE 2.2
Mode of Revolution
of Rotor Axis
Pitches
Banks to the left
Dives
Banks to the
right
Mode of Deflec-
tion of Cone
Forward and left
Right and
forward
Backward and
right
Left and back-
ward
Mode of Revolution
Mode of Def 1
of Rotor Axis
tion of Con
Dives and banks to
Backward
the right
Dives and banks to
To right
the left
Pitches and banks
To left
to the right
Pitches and banks
Forward
to the left
The deflection of the cone during rotation is easily determined if we re-
call that it is deflected to the side opposite to the rotation (in the plane of
rotation) and, furthermore, is deflected in a perpendicular direction opposite
to the direction of the gyro reaction (a right-handed gyroscope, on sloping back-
ward, is deflected to the right while the additional slope of the cone occurs to
the left).
5. Physical Meaning of the Obtained Result
Equations (2.41) show that, in response to rotation of the rotor axis to
some side, the axis of the cone in this direction will lag (the plane of the
blade tips, relative to the shaft axis, is^ deflected to the opposite side) and,
furthermore, will incline in a direction perpendicular to the rotation.
It is not difficult to demonstrate that the lateral inclination is caused
by aerodynamic forces and the lag by inertia forces.
As a typical exanple, let us examine the case p, = 0; B = 1.
Let us assume that the rotor revolves backward; Qy > 0; Q^ = 0. Then, /71
according to eq.(2.24),
t/y = X/?u) - r -^ - lj./?p cos '> + Qyr cos ,<p = Uy^^ + At/,,
where Uy is the esqaression for Uy when n = 0.
The angle of attack of the forward blade sections (i|r = rr), assuming \ =
= const, decreases by
A{/y
while the angle of attack of the blade sections stationary with respect to the
flow increases by the quantity
70
Under the effect of the change in moments of aerodynamic forces occurring
in this case, the flapping motion of the blades mil change until a new equi-
librium of the moments and a corresponding flapping motion are established. As
follows from the condition of zero hinge moment, the angles of attack of the
blade sections should retiirn to their previous value corresponding to recti-
linear motion (since no new forces appeared and the velocities U^ remained
constant).
This is realized when the real axis of rotation of the cone swept by the
blades is deflected leftward by the same angle Ao? (the axis is inclined to the
right to reduce the angle of attack of the advancing blade in the case of regular
stationary flight) .
Actually, the formula for bi Ceq.(2.39)] will yield in this case:
i.e., the slant of the cone to the right, occurring in rectilinear flight, de-
creases by the indicated quantity.
Now let us examine the effect of inertia forces. The additional force act-
ing during the rotation is the force due to Coriolis acceleration. Its moment,
accoiTding to eq.(2.7), is
A1
^= — f ^rnQyinr"^ slw!^ dr = — 2/^ ^ QyU sirn}».
i.e., during backward rotation of the rotor the advancing blade ( \|i = -^r-] is
acted on by the moment M^ = 2lh.h^yU^ which tends to depress the blade.
This new moment changes the blade flapping. Equilibrium is established
when, as a consequence of this flapping, the angles of attack of the blade sec-
tions in a forward position increase so that the additional aerodynamic moment
equalizes the moment of the Coriolis forces.
An increase in the blade angle of attack at the position i|f = -— - is achieved
when the real axis of rotation of the cone described by the blades is deflected
forward (imder rectilinear flight conditions, the axis is inclined backward so
that the angles of attack of the advancing blades diminish) .
Actually, eq.(2.38) for ai yields Z22
, 8Qy
71
i.e., during backward rotation of the rotor, the cone follows the rotor axis
So
with some lag, being deflected forward from it through the angle ^
yw
6. Equation of Torque
The elementary moment due to aerodynamic forces can be written in the form
of eq.(2.26):
dM^= 1- bQ {c.p^Ul - a^^U,U, - aMl) r dr.
Here the braking moment is considered to be positive. In addition to these
forces, additional inertia forces [eqs.(2.5), (2.8), and (2.10)] appear in the
plane of rotation owing to rotation of the rotor axis in space.
The expressions U^Uy and Uy in curvilinear motion can be represented ty
eqs.(2.23) and (2.24) in the following form:
t/fAd/y= — 9_^u)[),/?r sln^ i]j — Q^^ior^ sini}) -|- (2.L2)
-j- Qycor^ cos i)j -|- 2yU)[j,/?r sin ijj cos ijj;
^Uy = — Sj-r sin tji "1- 2yr cos <});
hUl = — 2Q^m\Rr sin ij; 4- 2 -?- Q^.wr^ sin <]> +
-J- 2p9^u)ij./?r sin tj) cos I}) + 22yO)X/?r cos i|) —
— 2 -^ Qy(or2 cos ij) — 2^QyW^Rr cos^ <]; —
— 22^2yr2sin6cos<J) + 2>2sjn2^_j_Q2r2cos2<;>. (2.43)
Then, transforming the expression dM,. and adding the moments due to inertia
forces according to eqs.(2.5), (2.8), and (2.10), we obtain
where
dM^=dM^^^-\-AdM^
* ^st ^ inert '
AdM^ = _ -i- 6Qa„ i<fU^MJy + A^P rdr-\-
-\-m
.22yiisin^-22,^cos^+(Q^^- 2^) sin <p cos ^^ - ^2.44)
~2^^.y^dr.,
72
BR R
t trf^J t«ro^J * inert (3.45)
Assuming p = ao - ai cos i|( - bi sin i|r and substituting into eq.(2.44) the /73
expressions U^AUy and AUp from eqs.C2.42) and (2.43), we obtain, according to
eq.(2.45).
BR
— tfQyiur^ cos tjj — tp2ya)(j,/?/-2 s jntj, cos ([) -j- 22_jU)X/?H sin t}i —
— 2a,2^<or3 sin^ i]j + 26iQ_f cor^sin tj> cos i]; — 2 (Oq — a, cos <]> —
— ftj sin i]j) 2^iop,/?r2 sin ijj cos i]) — 2Qy (oX/^r^ cos tj) -|-
-|- 2a,2y(Br^ simp cos <); — 2&i2yU)r^ cos^ tJ) +
+ 2 (flg — a, cos tjj — di sin <J>) Q^coix/^r^ cos^ tji -j- 22^2yr' sin tjj cos tj) -
R
— 2>3sin2tl)— 2^r3cos2<|j]a(r+ f m[ — 2ai2yCosin2.j)-|-
-j- 26,2yU) sin (J) cos tj* -[- (Qj/ — 2^) sin i}) cos ^ —
— 2(ao — Oi cost]) — 6isintj()(ai sini]> — A; cos tjj) co^] r^ rfr.
The moment due to z^ of the blades per revolution is
^t = ^ J ^1 ^'l- = ^ t .t + ^ ^*ea„-R*'o^x
X
r53 Q_^ fi4 Q_^
r0» i.!^ i(< w^ ±(4Uy
L D o) 4 a) 4u)
+^--Te)"-T(^;]+
(2.46)
Qj. Q«
Here M^ is the moment due to aerodynamic forces, obtained in the theory
for rectilinekr motion
^t st= '"trt4"'^''*^^''"' = '"trt T""^ ^'"^^^ ^^'
Ht ^lia^ ^ ^^ ' 3 ■ 2 8 I, ^ 2 '^ ; (2.47)
B2 „ B3 ^2 / B2 \ ;,2 1
73
We then write the expression for nit ^7 means of eqs.(2-46) and (2.4?):
r Cj-p 53 B2
.2
-^(i<'+Asv)-f»>=+f».V-
+-r W 7 — «!--; f'i + -^ — ^oH'-
D (1) 4 (d 4t»> 0(i>
■ey-TCfy+T(v^-°v«.)■
7- Rotor Thrust and Angle p.f Attack /74
The rotor thrust is
Substituting the values of U^ and Uy from eqs.(2.23) and (2.24) and inte-
grating, we obtain
Setting
T=l-toQnR*w-^ = ial~Q{wRyF, (2-49')
we find
L 2 ^ 3 V ^ 2 ^ ) 4« n (2.50)
As we see, the expression for the thrust coefficient has been somewhat
changed; nevertheless, assuming ^ = 0, we obtain the same expression as for
the case of rectilinear motion. The formula for the angle of attack, after de-
termining the induced velocity, is obtained from the expression
r = 2jt/?2QV"i>, (2.51)
where V is the resultant velocity,
V' = Y{Vsina-vf + V?cos^a. (2-52)
Substituting T and V by their expressions and remembering that
74
K sin a — v K cos a
" V---
we obtain
. X , at
*" ~~~^'^ 4h./X2+7^' (2.53)
i.e., the usual e^qDression for the angle of attack in which only t has a new
value. Writing this in expanded form, we derive
,_i,--[T-iKt")-^n
,„«=--r. - ^^-=^ ■-. (2.54)
8. lateral Force
According to eq.(2.28), the elementary lateral force is
We can represent dS as
dS= icostjj — ^dTsin<^.
ciS-=dS^^-\-^dS,
where dS^t is the elementary lateral force for the case of rectilinear motion:
AdS^-^^^cos'^-^AdTsiTK^. (2.55)
In eq.(2.55) we can replace AdMt = AdM,.^ ^ since the inertia forces /75
when summed with respect to z^^-blades will give zero. It is easy to demonstrate
this by means of eq.(2.44):
io^ COS <{<:;= — l™«£Lsin-j^ = 0.
u
Then, losing eqs.(2./i4), (2.42), and (2-43), we obtain
ArfAft..„ J
cos '> = — bQa„ [cp2_^(i)(j./?r sin^ i}» cos tjj -f
r 2
-|-tf£2_^u)r2sintJ;cosijj — cpSyfor^Cos^tp — tpQy(n[ji^rsin<l)cos^i}i-j- (^ rz'v
-f 22^u)X/?r sin i); cos ([) — 2a,Q_f(Br'^ sin^.ij) cos 4> +
-|- 26,a^a)r2 sin ^ cos^ t|) — 2(ao — a, cos ip — 6, sin t|j)Q^o)|x/?r sin <J( cos^ tjj —
— 2Q^mKRr cos2 ij)-f 2ai2j,a)r2 sin <!/ cos^ <]; —
75
— 2&iSy(i)r2 cos^ ij> + 2 (Co — a, cos <!) — frj sin ^) Qy<B(j,/?r cos^ <!/ -i-
+ 22^Qyr2 sin •]> cos^ <]; — 2 Jr^ sin^ <{) cos <!> — Q^r^ cos' <];] dr.
Setting
we find, by means of eq.(2.42),
pA rfr sinif) = — bQa^ (a^ — a^ cos tj> — d^j sin ij;) ( — 2^o)|i/?r sin' <!> —
— 2,(0^2 sin^ ijj + 2y0)r^ sin ip cos <]j + 2yO)|jL/?r sin^ (j> cos i})).
The lateral force of the rotor can be represented in the form
5=5,t+AS,
where
2<i B/?
a5=-^ ?^cos<J-flfrar.j;-
2r. BR
~lrj f?Aa^rsin<J;flrrc?<|-. (2-59)
(2.60)
Substituting eqs.(2.56) and (2.5S) into eq.(2.59) and integrating, we
obtain
Substituting S^t ^7 its value of /26_
+^(B=X+^a.,.)]— |-aoi.(m+^cp)+ (2.6I)
we finally obtain
76
+-^t(i— t*''^)+-;7t(t+'+t"'^)J- (2.62)
9 . Longitudinal Force
According to eq.(2.27), the elementary longitudinal force is
or// =^^sin <); - p orr cos (|>.
Setting
we find
AdMf
^dH= — ^^sin.]>--PAafrcos<l;. (2.63)
Analogous to eq.(2.56).
*-oxro . 1
•slni]) = — bqa^
(f)Q_j.o)[x/?r sin^ tjj -j- tpQ^mr^ sin^ <ji
r ' 2
— cQ^mr^ sin ijj cos <]> — cpQ„(u|j,/?r sin^ 4* cos (j) -\- 22_^u)X/?r sin^ li —
— 2ai2_^o)r2 sin't}) -f 26iQ_jO)r2 sin^ <h cos (]j —
— 2 (^o — a, cos (]< — ij sin ijj) 2^o:[j,/?r sin^ t); cos <J) — ( 2 .64)
— 2QywKfir sin <]; cos iJj + 2a,2yU)r- sin^ ip cos tjj — 2bj^Qywr'^ sin iJj cos^ <{; +
-j- 2 (ao — a, cos i]j — &i sin tjj) QyO)(j./?r sin tj; cos^ <^ +
+ 2 Q_^2yr2 sin^ i|) cos tj^ — Q^r^ sin^ ;|j — Q^r^ sin ijj cos^ t{i] c?r.
Analogous to eq.(2.58),
pA ofr cos ijj= — - ftga™ (flg — a, cos <]j — Aj sin (jj) ( — 2^(fl|j,/?r sin^ ti cos
— S^iur^ sin ij> cos ijj + Q^utr^ cos^ '^j + 2y(oiJ./?r sin ij) cos^ <])).
(2.65)
Let us use the same notations as in the calculation of the lateral force,
namely
2it BR 2ic BR
^^^^^^ '—^'^^^^'■d^-f^fj^^drcos^drd^. ^^'^^^
Substituting here eqs.(2.64), (2-65) and integrating, we obtain /77
77
2
B3_
6
2a ^ ^=P>-|*+-V'P^i
(2.68)
The e25pression of the longitudinal force for rectilinear motion has the fonn.
Summing eqs.(2.67) and (2.68), we finally obtain
+T*+f-.^+f »>+°t ^iCf +^+T^-)- (,.69)
-TT(°f+iv)]-
10. Consideration of the Chan&e in the Law of Induced
Velocity Distributipri du ri ng Curvilinear Motion
In cujrvilinear motion, a change both in magnitude and in character of the
induced velocity distribution over the rotor disk should occur owing to the
presence of new forces, namely Coriolis inertia forces; the moments produced by
these forces are balanced (in the case of hinged blade attachment) by the redis-
tribution of aerodynamic forces.
It is logical that, if the angles of attack of the blade sections at a
given angular position in curvilinear motion do not change in con^iarison with
those existing in rectilinear motion, then both the forces and the induced ve-
locities caused by them also will remain constant .
Assuming that the rotor revolves backward (fiy > O), the angle of attack of
blade section, at iJ, = 0, B = 1, will be
Uji ^/?w — rutoi sin ij; + rmbi cos iJ' 4- Syr cos <}'
Substituting here the e^qjressions for ai and bi obtained from eqs. (2. 38)
and (2.39) and setting p, = 0, B = 1, and Q^ = 0, i.e..
we obtain
8Qj,
^ =
2y
Y™
(1)
X/? 82},
ar=<p-j s'm<]i.
r •yo)
The angles of attack in the forward and rear positions (^ =0, i|r = tt), as
shown above, do not change when \ = const, i.e., the kinematic change in velocity
78
of the disk (flyV cos i|») does not produce a change in the angles of attack of the
blades thanks to the corresponding change in flapping motion.
The above statements also hold for p. ^ 0. The curves of the azimuthal /78
change in angle of attack of the blade section, plotted in Fig. 2. 23, for recti-
linear motion and for rotation of the axis in a longitudinal direction also show
that, at azimuths t = and ijt = it, the angles of attack, for all practical
purposes, remain constant*.
The change in angles of attack, for the exanple under study, takes place
from the right and left of this position, the maximum change occurring when
^ = -S- and ^ = -i- TT, which corresponds to the azimuth of the maximum value of
the Coriolis inertia forces.
Thus, it can be assumed that the previous distribution law of forces and
induced velocities over the disk is si?)erposed by aerodynamic forces equalizing
the moments due to the Coriolis inertia forces and the resultant velocities.
These forces have a maximimi in a plane perpendicular to the direction of rota-
tion, so that the induced velocity field will be tilted in this direction.
The vertical conponents of the Coriolis forces are ejqjressed, according to
eq.(2.7)> in the following manner:
R 1
Ks = — ("2m2j,(orslniJjrfr= — 25j,^,Qv"'sintjj; 1
R
ATa = — f 2mQ^<f>r cos <\)dr = — 2S^^^Q^<» cos ip.
(2.70)
These forces are linearly distributed over the radius and are periodic.
It can be assimied that the aerodynamic forces and their induced velocities,
equalizing the Coriolis forces at each angular position, obey the same law of
change both with respect to azimuth and radius of the blade. Then the induced
velocity in the case of curvilinear motion can be expressed in the form
i>==t)o + '"i-^sirnj.+t;2-^cos.l). (2-71)
Here, the velocity directed downward is considered to be positive. Let us
denote
Py = =C
vo T
Px= =C '
(2.72)
■«- The difference in the angle of attack of the section, at \|( = 270° , for rota-
tion in a transverse direction (see Fig. 2. 22) is due to a change in X .
79
Here, I Kq I and I Kq I are the absolute values of the CorioHs forces at
angular positions where they reach a maximum. In calculating Vq, it was assumed
that this velocity is constant over the radius. Assuming an additional induced
velocity as a linear function of the radius, we should, from the condition /79
of equality of momentum, introduce some factor C into the esq^ressions for p^
•a
and py . At C = —-
Pr
3Sv,.w^b 2y
1 '
— aonRH
(2.73)
Substituting Vx and Vg ty their values from eqs.(2.72), we obtain
R
R
We then e^qiress the mean induced velocity in terms of the magnitude of
thrust. According to the law of momentum
27t/?2 qV
(2.74)
Here V' is the resultant velocity determined from eq.(2.52). Substituting the
expressions for T and V' from eqs.(2.49) arid (2.52), we obtain
_ QtlS,R
Let us denote
X,=
f
it
^R 4/X2 + H-2
Then, the velocity conponent Uy of the blade element will have the form
(2.75)
f/' =X/?u) — /7 Xircoslntj)— p jXirw cos <]i~r — —
dt
■ |i,/?<Bp COS <{> + 2yr COS i]> — Qj^r sin i[).
(2.76)
To the expression U^Uy obtained for v = const, we then add the terms
UjiA'Uy = — jOy^^ir^to^ sin (Jj — pyXj<fl2/?/-ix sin^ i)j —
— pJ^irV cos ({j — Pj^i'o'^Rr^ sin i]j cos tj».
(2.77)
The expression for the rotor thrust will have the form
80
in BR
r=^J J -|-&Qa-(9t/^ + t/,d/y + t/,A't/y)rfr-rf1',
which, in the esspression for the coefficient t, will give the additional term
In the expression for the thrust moment, we obtain the additional terms
BR BR
Substituting the expression for the thrust moment (with these additional /80
terms) into eq.(2.30), we obtain the following expressions of the flapping coef--
ficients, with consideration of a variable law of induced velocity distribution:
a, = 2,(x+-i- 5cp) L^ (fi^"l-+«-^+5V/,)
B2+-i-^2 ^ - y^ ^B2(b2_J-^2)' (2.79)
1 \ MVM / I \
Let us now derive the formula for torque in the case of a variable law of
induced velocity distribution.
We find the expression for Uy. Let us denote
A'i/y = — p^irui sin (p — /t7^>-iru) cos <1>, ( 2 .81)
where Uy is taken with respect to eq.(2.24), i.e., without consideration of a
variable induced velocity.
Then, , , o
= U\ — 2kpykyR'r sin ^ — 2lp^^<i>'^Rr cos if +
81
+2 J^Py^yr^^in<^-\-2 ^p;kyr^ cos^+
+ 2P)PyXj{j,a)2/?r sin <j< cos <)» + 2^pj^^^m^Rr cos^ .ji _ 2Qympy\r'^ sin <p cos <}» +
+ 22^a)Pj,Xir2 sin^ <]) — 2a,(i)yo^i/-2 cos^ <}. + 22^(j)/?^i sin tp cos <J> +
+ /7^X2a)V2 sin2 .p + 2/»yP^y r^ sin (|) cos >}» + /72 X^w V cos^ <}..
The expression for dMt reads
dM[ = y-6e ('^x;,^,^^ - a„^U_,U'y ~ a„ U'y) r dr.
We Ccin represent this in the form
- fl» (Ul + 2t/yA'(/y + A'^p] f dr
or
where
A'rfA/t = - ^6oa„ (?A't/^^y + 2t/yA't/y + A'f/prrfr. (2.83)
Substituting here the necessary e25)ressions from eqs.(2.77) and (2.82) /81
and integrating, we obtain
2k BR
00
or, setting mt = mt + A 'nit, we obtain for A'mt the e35)ression
We now determine the expressions for PxXi and py\i with the aid of
eqs.(2.73) and (2.75), assuming X to be small in ccaiparison with |j, (which holds
for jx s 0.15):
(2.85)
82
3 Su L j-L 2-
X — ' ~^^ J^ !^
2 Q3l/?4 H
(2.86)
Obviously, the additional terms in the expressions for the flapping coeffi-
cients and in the expression for the torque do not depend on X .
Equations (2.86) show that the smaller the value of (j,, the greater will be
the influence of distortion in the induced velocity distribution. This becomes
understandable if we recall that the smaller the translational velocity, the
greater should be the induced velocity caused by an aerodynamic force of the same
magnitude. The fact that the quantities Px^i and pyXi do not depend on X greatly
facilitates the calculation of the flapping coefficients and forces from the
variable induced velocity. The calculation in this case is performed in the
same manner as for a constant induced velocity, except that ao , a^ , and bi are
calculated from eqs.(2.78), (2.79), and (2.80) and the expression for mt from
eq.(2.48); to the obtained value, we add the term A'm^ calculated from eq.(2.85).
ANALYSIS OF OBTADJED RESULTS
11. Blade Flapping
To illustrate the point, we made various calculations of a rotor in curvi-
linear motion.
Design data:
rotor diameter, D = 14 m;
loading coefficient, a = 0.065;
static moment of blade relative to axis of flapping hinge, S^.h =
= 1^1-2 kg • m;
angle of blade setting, cp = 3°, a^o = 5.?.
All calculations were performed for the regime of autorotation.
Figure 2.1? shows the variation in flapping coefficients, calculated for
a constant value p. = 0.3 and different values of the ratio ^ (rotation in /82
transverse direction). We see from Fig. 2. 17 that, on rotation to the left,
{ — 5— > q) bi increases while ai decreases, i.e., the cone described by the
blades is deflected to the right and forward. In rotation to the right
f — s_ < o), we ha,ve a decrease in bi and an increase in a^, which indicates de-
flection of the cone to the left and backward. The coning angle ao slightly in-
creases with rotation to the left and decreases with rotation to the right.
83
Consideration of the change in the induced velocity distribution leads to
an even greater change in bi at an angular velocity Q^.
Ciorve bi calculated for a variable induced velocity is plotted in Fig. 2. 17
as a broken line. For
practical purposes, we can
ao.i^i.bi.flmln
-
„
-
S"
■3%
l<
/
/
-
^
/-
^
S^ J
flo
i—
""
/
=-
^
■r-'
u
/
^
S
/
"-^
-
//
//
/
b,
—
/
/
/
Si
\
■<
"^
—
p>min
(^
^
r
L
>
^/
"V
■n
\
N
s
^
/"
^
^'
-0.
oiy
^0.03,
'^0.02 -0.01
0.01
0.02
O.L
7J
0.0^
El
((I
/
y
"■
40
--
V
/
/
/
1
- -z'
-.7"
-
/
'/
/
/
-
/
disregard the changes in ao
and ai due to variable induced
velocity .
Figure 2.17 also gives
the curve for the values
Pml n ■
shows
Pmln
— vai
^2
which
a marked decrease in
for rotation to either
Fig. 2. 17 Flapping Coefficients in Transverse
Rotation of Craft (iJ. = 0.3).
left or right. This means
that the reserve of blade
overhang decreases markedly
toward the lower arresting
device. The blades will pass
lower, the greater the angular
velocity of the roll and the
smaller the value of y •
Figure 2.18 gives- the
same curves for the case of
rotation of the craft in
longitudinal direction. In
conformity with the foregoing,
diardng rotation in a dive, ai
increases and bi decreases,
i.e., the plane of the blade
tips is displaced backward
and to the left, whereas during rotation in pitch it is shifted forward and to
the right. Curve Pnin shows that nose-down rotation causes the greatest de-
crease in p„i^, i.e., the blades pass very far below the plane of rotation,
whereas pitching rotation in this sense is most favorable.
M
Figures 2.17 and 2. IS pertain to craft whose control is not acconplished by
means of an automatic pitch control mechanism (for exanple, by ailerons and
rudders, by deflecting the hub of autogiros with direct control). Both
Figs. 2. 17 and 2. IS indicate that, for such craft in curvilinear motion (for
exanple, during a sharp bank or going into a dive), the change in P„i„ may be
much greater than its variation over the entire speed range of the craft in
rectilinear flight. This should be taken into consideration in selecting the
position of the lower arresting device of blade flapping. On a helicopter, the
pilot, in deflecting the automatic pitch control, reduces the deflection of the
blade cone in curvilinear flight; thus, the reserves of the angle toward the
lower arresting device do not diminish so severely.
84
Oo,a,,b,,li„i„
■^
X
<
.b,
—
7*
^
,y
^
6"
5'
^
-
"o
y^
<-•
N
\^
,
^
^
y
y
\
^
-
/
3%
2°
<
A
-
\
N
/
'Pmin
/
■-•-
^
■^
N
>v
X
7
/
■^
;n^
^
■^
—J
0.05j-0.0t
-0.03
-0.02,
'-0.01
0.01
0.02
0.03
ao^
■\
/
1
-1°
/
/
r>o
-
</
-'
-
tO
/
/
1
/
f
_
1
-J"
1
783
«2
Fig. 2. IB Flapping Coefficients in Longitudinal
Rotation of the Craft ((jl = 0.3).
"fai
:i. I 1 I I iTi
Fig. 2 .19 Change in Angle of Attack and Thrust Coefficient
of a Rotor, as a Function of the Angular Velocity of
Rotation of the Craft at Constant iJ. = 0.3 •
m
85
12. Effect of Curvilinear Motion at Autorotation
of the Rotor
In the aerodynamic, calculation of a rotor in autorotating regime, eq.(2.48)
is used for determining X, -which permits finding the value of X, if mt = is
assimed and cp, iJ., CI,., and Cly are known. However, in practice this quadratic
equation becomes very cumbersome after substitution of the values of ao, ai, and
bi. For determining the value of X it is more convenient, after plotting the
dependence of m^ on X, to read from the graph the value of X at which mt van-
ishes. When constructing the plot of mt, the coefficients ao, a.x, and b^ should
be determined from eqs.(2.37), (2.38),
and (2.39).
K-gure 2.19 gives the values of /84
the angle of attack of the rotor and the
thrust coefficient t, obtained for a
constant value y. = 0.3 at different
angular velocities of rotation of the
craft .
Figure 2.19 indicates that, to
maintain a constant value of |j, at con-
stant flying speed in the presence of
angular velocity, the magnitude of the
angle of attack must be changed and the
more so, the larger the angular velocity.
When the craft rotates to the left, a
constant value of ij, can be maintained
only by increasing the angle of attack
over that in rectilinear motion, whereas
in rotation to the right the angle of
attack must be reduced. This is ex-
plained physically by the fact that, in
rotation to the left, the rotor disk is
Inclined forward (in addition to being
inclined to the right), which reduces the angle between velocity and plane of
rotation of the blade tips in space. In rotation to the right, the opposite
occurs. It can be asstimed that, to maintain a constant value of [j., the angle
between the velocity of flow and the plane of rotation of the blade tips must
remain constant at any angular velocity Q.
Figure 2.20 gives the values of the angles of attack as a function of |j,,
obtained at different angular velocities of rotation of the entire craft. It
can be seen that, if rotation of the axis begins and proceeds at a constant /85
angle of attack (i.e., angle between velocity and a plane perpendicular to the
rotor axis), then steady autorotation will occur only at a new value of p,, dif-
fering from the former. Thus, in rotation of the craft to the left at a constant
angle of attack iJ. will increase and in rotation to the right, it will decrease.
If in addition to the angle of attack also the flying speed is kept con^
stant, the rpm of the rotor will decrease on rotation to the left and will in-
crease on rotation to the right.
a"
-
\
\
S
^
-
\
fS
L
\
\
V
\
\
10
\
\
\
\
\
V
\
\
\
\
\
^ = + 0.05
s
\,
\
\
\
Vl
>,
\
N
^
\
N
. —
&
=n
^
^
\
""V
->«
*"
"i
N
11
0.2
C3n
.
0.1
^
A
52,
■ — n/iK
^
(ti
I — 1
—
—
Fig. 2. 20 Angle of Attack of Rotor
in Autorotation.
86
To illustrate this phenomenon. Fig. 2. 21 gives the revolutions of a rotor at
angles of attack of a = +4.5°, 2.8°, and 1.5° ((io = 0.25, O.3O, 0.35 in recti-
linear motion) as a function of the velocity of rotation in transverse direction.
The number of revolutions in rectilinear flight is taken as no = 200 rpm.
The rpm in the presence of rotation
nQ is obtained from the following
considerations :
=9,55
Ks=9.55
Kcos g
Vcosa
-0.05 -COi^ -0.03 -OM
Fig. 2.21 Rotor Revolutions in Auto-
rotating Regime, as a Ftinction
0.
of
(U
ras = /to — ■
The value of i^q is determined
from Fig. 2. 20 at the intersection of
the ordinate 01 = const with the curve
a = f(iJ.) for the corresponding angular
velocity .
As a consequence of the above
effect of an increase in angle of
attack necessary for maintaining
steady autorotation at a given |j,,
there exists the risk of the rotor
losing autorotation at high flying
speeds in the case of rotation of the craft to the left.
If the helicopter begins to rotate sharply to the left at large \i. while its
angle of attack remains constant, the rotor revolutions will decrease rapidly
and |jb will increase further. This is aggravated by the fact that the forward
inclination of the resultant of the rotor, generated when the craft rotates to
the left, will per se (against the will of the pilot) create a diving moment
which tends to decrease the angle of attack even more. The latter circimistance,
increasing the abrtptness of drop in rpm as the craft rotates, involves an in-
crease in angles of attack with respect to the blade sections, which causes flow
separation and marked increase in drag.
Figiores 2.22 and 2.23 give the variation in angles of attack, calculated
for sections of r = 0.5 at p, = 0.3, for the case of stationary flight and for
different directions of rotation of the craft. As seen here, the angles of
attack with respect to the sections increase markedly on left-hand rotation and
nose-down of the helicopter. This is due to the fact that, in the cases under
consideration, an increase in angles of attack at angular settings of the blade
of i|( = 270° and i|t = 180° .(where the angles of attack of the sections are already
large) is required to balance the moment of the Coriolis forces.
Taking the variable Induced velo'city into consideration will always in-
crease the variation in angles of attack with respect to azimuth.
87
a
^'r
r~0.5; fi=0.3
7
r-
\
Rotation to left
-ff=+m
/
\
IS
/
^
/
\
f
i
\
Bo
tation to ri
iht
.
-4
/
\
\
^ = -0.05
1
y
\
\
w
1
/
\
\
/
j .
I-
\
\
< Without rotation
X - — 1
/
J
1
\
\
\
\
\
9=0
'~~-
n
/
j
\
\
1
/
1
I
-
s
\
/
/
/
\
\
\
J
/
/
\
N
\
/
/
\
"s
N
\
I
/
/
\
A
\
/
\
— — .
—
^>.
^
—
100
200
300 360 (f'
Rig. 2. 22 Angle of Attack of Blade
Section vs. Azimuth.
oi"r
"TTl i 1 i
1 1
7\
—
^
i
«
—
Roto
tion in diving — '
1 '^y- nm
\
w
1 _. I
tation —
—
—
L
^
^
\
V
-
—
Without ro
V
W
-
LIJJ
Pitching
-
-
i
y
—
N
'
N,
^y ^n/T!
[h
^
\
>-
^
■^
1
\
\ —
^-
—
//
—
f^
n
^
5
—2
-^
-
y
/
P
—
—
V
-N
■v.
^
-
—
"
—
—
^
V
^
100
/
2-00
---
3UU
3tL
H
\
\
—
z^
L
_
1
Pig. 2. 23 Angle of Attack of Blade
Section vs. Azimuth.
Based on vdnd-tunnel tests /86
of blades with two profiles, of
which profile Mo.l has higher /87
values of c,
and O'er than pro-
file No. 2, we plotted the curves
of the reserve of autorotation in
Fig. 2. 24 (Ref.ll):
a — Qq— tan" -^=/(a — Oo),
Cy
where a^ is the angle of attack
at which Cy = for the profile.
The diagram shows that, in the
case cp = 3°, the range of angles
of attack, at which an accelerat-
ing moment is produced on the
blade element extends from 0?^ =
= 4° to ffr = 15° for the profile
No. 2, and is much greater for the
profile No.l, reaching a value
of 30°.
In profile No. 2, during left-
hand rotation of the craft with
a ratio of
Q,
CJU
= 0.05 at n =
= 0.3, a decelerating moment will
he produced even on the section
r = 0.5, over an appreciable por-
tion of the disk. To maintain
autorotation at given |j,, an even
larger angle of attack of the
rotor is required; if the angle
of attack is kept unchanged, the
rK>tor rpm will drop and the flap-
ping motion will increase greatly.
This brings us to the conr-
clusion that a constant (accoiKi-
ing to Fig. 2. 21) decrease in
jTotor rpm on left-hand rotation
of the helicopter and an increase
in rpm on right-hand rotation
will take place only up to some
small value of
U)
whose magni--
tude is determined exclusively
by the aerodynamic blade charac-
teristics. At large values of
88
A-dg- tan-' ^
o.
U)
Fig. 2. 24 Curves of the Autorotation Reserve of Blades,
the rotor rpm on rotation to the left will drop more abniptly - even as far
as loss of autorotation - as a result of flow separation, whereas in rotation to
the right the rpm vri.ll first cease to increase and then, at large values of
U)
-, begin to decrease.
Thus, for a rotor with blade profiles of large autorotation reserves one
can safely permit a much greater angular velocity than for a rotor with poor
blade profiles.
13- Behavior o f the Re sult ant of Aerodynamic Forces
dji Cur vilinear HeUicopter Motion
/88
According to general considerations, the resultant in the case of curvi-
linear helicopter motion is deflected in a manner similar to the deflection of
the cone described by the blades in space. Thus, the resultant lags on the side
opposite the rotation and, in addition, is deflected in a perpendicular direc-
tion by an amount proportional to the ratio
The lag of the resultant causes, relative to the center of gravity of the
craft, a moment counteracting the rotation. This constitutes a damping moment
which is larger, the greater the angiilar velocity of rotation of the craft.
The inclination of the resultant in perpendiciilar direction in the case of
lateral rotation causes a change in angle of attack, whereas in the case of
longitudinal rotation the inclination of the craft is to the right or to the
left.
Figure 2.25 gives the variation in the quantity -7=- characterizing the
angle of inclination of the resultant in the lateral plane, as a fimction of the
89
s
1
0.07
0.06
0.05
0.01
0.03
0.02
0.01
/
-
-
/
/
/
/
/
/
/
0.05
-mv -0.03 -0.02 -0.
^.OZ
-0.03
-0.0V
-0.05
-0.05
-0.07
-008
0.01
0.02
0.03
0.01
&
/
/
/
/
/
V\
Fig. 2. 25 lateral Inclination of
the Resultant as a Function of the
Velocity of Rotation in Transverse
Direction.
of the rotor with respect to the degree
cients .
angi^Lar velocity of banking — s_ (at
CD
p, = 0.3).
The formula for the lateral force
in rectilinear motion does not yield
the values of S close to full-scale
values (owing to the poor convergence
of the calcxilated flapping coefficients
to the real coefficients); however, the
variation in lateral force as a func-
tion of the angular velocity is cor-
rectly given t^r eq.(2.62).
The slope of the curve -^ =
n.
cu
curve bi = fl
■j is close to that of the
This circumstance
can be used for an approximate calcula-
tion of the danping forces and moments
of variation in the flapping coeffi-
The magnitude of the conponent of the danping moment acting in the plane /89
of angular velocity of the craft is readily determined from the formula
M = Tby.
where
y = distance between center of hub and center of gravity of the craft;
6 = angle of lag of the resultant in rotation.
According to eqs.(2.80) and (2.86), we can take, for n s 0.2
(0 \
Y 2 Qrt/?4(i )
B2(S2+-|-|.2)
K,.
Here, K^ is a coefficient taking into account the change in the flow coef-
ficient \ during rotation of the craft. The coefficient K^ can be determined
from experiment. If no e35)erimental data are available, we can take K^ = 1.
The intensity of interaction, expressed by the slope of the curve S/T as a
function of ^^M, may decrease on further increase of this ratio above a certain
value, owing to flow separation in the blade sections.
As a result of the above-described phenomenon of decrease in rpm and hence
in thrust during autorotation while the craft rotates to the left (with a rotoi-
of right rotation), the damping and thus also the control 1 ability of the craft
is less in the case of left-hand rotation than in the case of right-hand rotar-
90
tion.
The use of an optimum supporting blade profile extends the permissible
range of the angles of attack of the blade sections so much that, with proper
arrangement of the helicopter, the critical magnitude of the angular velocity of
rotation of the helicopter will not be reached in actual service.
EFFECT OF HDTOR PARAMETERS AND HUB DESIGN ON FLAPPING AND
DAMPING OF THE ROTOR
Blade flapping and deflection of the resultant in ciorvilinear motion of a
helicopter are affected by the characteristics of the rotor itself, which
"Changes its aerodynamics under these conditions.
Below, we will examine the effect on flapping and danping of the rotor, the
moment characteristics of the blade profile, its transverse centering, and hub
design.
Let us take a hub with a flapping con^^ensator, with kinematic dependence of
the angle of blade setting on the flapping angle - such that the angle of blade
setting decreases with increasing flapping angle.
14. Rotor with a Profile , Having a Variable Center of Pressure
Until now, we discussed a rotor having blades with profiles of constant
center of pressure and with a flexural axLs coinciding with the center of pres-
sure. Let us now examine a rotor
having blades with a variable center
of presstire.
Recalling the variations in the
angle of attack distribution of the
blade sections with respect to azi-
muth as they occur in curvilinear
motion, it is easy to show that, if
the coefficient of the moment rela^
tive to the flexural axis of the
blade c„ depends on the angle of
attack of the section, then the
aerodynamic moment producing blade
twist will vary in relation to its
angular position. This, as a re-
sult of blade twisting, will cause
a change in the flapping motion and
in the position of the resultant.
Let us si:ppose that the flexural
axis is located aft of the aero- /90
dynamic center and that the profile,
at aj. = 0, has a diving moment
Fig. 2. 26 Diagram of the Effect of
Coriolis Forces Producing Blade
Twist in Curvilinear Motion.
91
(c„o < 0)» i«e«, vdth increasing a^ the center of pressure of the blade section
shifts forward and the diving moment c„ decreases.
As an exanple, let us examine the case of left-hand rotation of the craft.
The character of the angle of attack variation of the blade sections, for
rotation to the left, is plotted in Fig. 2. 22. For the blade in forward position
{•if = tt), the angles of attack decrease almost to zero whereas for the blade
located aft of (f = O) they increase appreciably. Therefore, the forward blade
is tvidsted in diving, i.e., the angle of setting decreases, whereas the rearward
blade is twisted very little (with the velocity U^ being identical in both posi-
tions) .
To balance the angles of attack of such blades, providing zero hinge moment,
the cone of the blades and hence the resultant should be deflected to the left.
This additional inclination of the resultant to the left, occiirring in left
rotation, decreases the resistance of the rotor to rotation of the craft, i.e.,
danping, and may cause the helicopter to bank at large angular velocities. This
effect produces pressiore on the control stick directed towa]7d the side of bank-
ing.
The above discussion shows that, if the flexural axis of the blade is lo-
de
cated in front of the aerodynamic center, i.e., if " < 0, the danping moment
of the rotor resisting rotation of the craft, increases.
15. Effect of Blade Centering
If we assume that the center of gravity of the blade section is located at
some distance bf from the focus of the profile (positive bf - backward), then
during rotary motion of the rotor axis a couple, produced by the Coriolis and
aerodynamic forces, arises on the blades. This is shown schematically in
Fig. 2. 26.
The ejqpression of the moment producing the blade twist (a positive moment
will twist the blade toward an increase in angle) will have the form
Substituting dKQ and dKQ by their expression from eq.(2.7), we obtain /91
M^={2b,m (2y(i) sin 1^ -|- 2^(o cos tj)) r dr.
It is obvious here that the moment varies periodically.
The angle of twist of the blade, \mder assunption of constant mass, a value
92
of t), , and torsional rigidity, is determined ty the formiila
C Mdr 2 bimRs /Q o >
Assuming that a linearly twisted blade is equivalent to a blade with a
constant angle of setting equal to the angle of setting of the first blade at
the section r = 0.75 R, we find
bcmR^
Let us derive an additional periodic variation in angle of setting
where
Av= Vj cos (j) -|- V, sin tj>,
Qv -
Vi=>^— ; v, = yi-^, >\=0,61
6j m/?3io2
The periodic variation in angle of setting leads to a variation in inclina-
tion of the cone during rotation.
The flapping coefficients take the form (for constant induced velocity)
1
-\-A
2 '^
^ ^ ' B'ilm+ — (i.2 j
-/I
B2 + — p.2
(2.87)
If the center of gravity is located aft of the aerodynamic center (bf > 0,
A > O), an additional inclination of the cone and hence of the resultant to the
side of rotation of the craft will occur. If the e.g. is ahead of the a.c,
then the additional inclination increases the danping moment of the rotation.
The angles of twist Vj, and v^ are easily determined if the dynamic twist of
the blade in rectilinear flight is calciilated and the angle of twist Vq is /92
93
knovm (Ref.6). The relation between these angles is determined by the formulas
2 Qy
(2.88)
V,=Vn
v,=v„
2 Q^
16. Rotor with Flapping Compensator
Let us now examine a rotor whose blades change in pitch cp, as a function of
the flapping angle p. There are many methods of acconplishing such kinematics.
Figure 2.27 shows one of the methods
of changing pitch in relation to the
flapping angle p (turned flapping
hinge), where cp varies in accordance
— . with the law:
Axis of blade
Axis of flapping
hinge
Fig. 2. 27 Diagram of Blade with
Turned Hinge.
to the case ij, > 0.
The flapping angle P, in this
case, is the angle between the blade
axis and the plane of rotation. Mot
wanting to conplicate the results, we
will take the case n = 0. The varia^
tion in flapping motion obtained for
regimes with p. = can be extrapolated
Let the craft be inclined in space at a constant angular velocity having
the conponents Q^ ^'^'^ ^y*
The velocity conponents of the blade element will take the form
U^=\wR-
■ r -6- + 2„r cos i]j — 2 ^r sin ip.
Assiming cp = 9o - P tan a^ , we construct, as above, the ecjuation of moments
relative to the horizontal (flapping) hinge for B = 1:
Y 2y
+5i+K'->-H(f+^)+
Y ^x
-j- -^ -^ COS t|j sinil) — 2 — sin--j< — 2 — cos^.
4 <0 4 0> * CO u
(2.89)
The particular solution of this equation has the usual form
p = flo — 'J'l cos <|> — &, sin (p,
94
while the coefficients of the series have the values
1
'^=v(i+4-)
1 -|- tonoj
4
2^
I u V Y / ' <o \ Y 7jl+t«2ai
L u) \ Y / 0. V Y VJl +t«A2oi
131
(2.90)
Equations (2.90) show that, in the case of a turned hinge, the deflection
of the cone described by the blades and hence the resultant will take place at
a smaller phase shift than in the case of a
conventional hub. The absence of phase shift
TABIE 2.3 means that, in transverse rotation, there should
be no change in longitudinal inclination of the
Y
8
6
4.6
0], deg
45
37
30
resioltant, i.e., at
Q
cu
ai = and, conversely, at
i^ = 0, we should have
ou
0, bj
0.
This condition is satisfied if
tanai=— r
(2.91)
The values of a^ , shown in Table 2.3, are derived with respect to the
value of Y •
TABLE 2.4
o, =
V
= 8
Y = 4.6
ai = 45°
o, =
oi = 30°
«!
_S3,
Qy
Qy
—1.74 —
Qy
- 1.74 —
(0
*I
Qy
Qy
1 *'
tan-^
«1
45°
30°
It is of interest that the condition of absence of phase shift tan a^ =
= y/8 yields the same magnitude of damping (lag of the resultant) as for a rotor
95
vd-thout flapping conpensator. For exanple, for longitudinal rotation we ottain
the values indicated in Table 2.4»
It is ob-vious that the absence of phase shift in transverse rotation at
constant angle of attack ensiires maintenance of jo, in the aut ©rotation regime,
i.e., prevents a decrease in rotor rpm or in controllability during rotation.
B3T0R FLAPPING IN CURVILINEAR MOTION OF THE ROTOR AXIS
AT VARIABLE ANGULAR VEIOCITY
Below, we will derive formulas for determining the flapping coefficients
of blades in the presence of uniformly accelerated and harmonic oscillation of
the rotor axis. For simplicity, we will take the case where |j, = 0. It is shown
that, for both laws of variation in angular velocity of the rotor axis, the /94
flapping coefficients and hence the longitudinal and lateral forces of the rotor
will vary by the same amoimt - proportional to the angiilar velocity - as in the
case of uniform rotation of the rotor axis. Furthermore, terms appear that de-
pend on the angular acceleration of the rotor axis.
17. Uniformly Accelerated Rotation of the Rotor Axis
Let us first examine the case of rotation of the rotor axis in pitching at
variable angular velocity'""
^> = ^^- (2.92)
Let us substitute t = -|- and put k = -^. We can then write
2y = % (2.93)
We then derive the eijqjression for forces and their moments relative to the
flapping hinge.
The velocities in the blade section, assuming v = const. |J. = 0, Hy = ktjf ,
Cl^ = 0, and B = 1, can be obtained from eqs.(2.23T and (2.24):
Uy=^y.i^R-i-k<]ir COS <l/~r -^ .
(2.94)
Then, the thrust moment takes the fonn
Af^=-i-j^»Qa„(cpt7J + ^^t/y)rc/r= (2.95)
^ By tiorning the coordinate axes through an angle \|fo , all results obtained below
can be extrapolated to the case of rotation in any direction.
96
2 \ 3 ' 4 4 rfij; ' 4(0 ^ J
The moment due to the Coriolis force is
M^,=---2/^^^k'^sln^. (2.96)
The inertia force of rotational acceleration reads
dQy
ajs=m r cos t|» dr,
where '^^
dQy d{ki/)
= •=«U).
dt dt
The moment of this force is
R
M]^= l"/?iAo)COs<})r2rfr=/^^^;feiocos'j;. (2.97)
u
The equations of moments relative to the flapping hinge, after canceling
by 1^,^ • (ju2, will take the form
Suu (2.9s)
The particular solution to this equation has the form
{)= flj, — a^'if cos •;) — i^i'^ sin ip — ^i cos tjj — afj sin ijj. (2-99 )
Let us find the derivative of p with respect to \|/ : /95
£L= ajijj sin 4' — a, cos (j> — 6,i]> cos ')' — &, sin ij;-!- c, sin 'J' — rfj cos t)!;
.£?L= flj,]) cos <})+ 2ai sin ([» -f frjij) sin i|j — 2&, cos <J<4- <^i cos '^ + ^1 sin iji.
dtj;2
We next substitute these values into the left-hand side of eq.(2.98),
yielding
-^a,<j>sln<]) —biifC0Si(-{-(2ai ^^iH — ^CJ^sintj) —
-(2*,+^H,+-Jrf,)cos.l> + «o = Y(|-+^)+ (2.100)
+ — cos <!> — 2 — <}» sin<j) + -^ — >}. cos <1< — -'"•^
<d CO 4 w /».
'v.v,"''
97
Since eq.(2-100) is an identity, then, hj equating the coefficients of like
terms, we ottain the following system of equations for detemiining the coeffi-
cients of the series
4 ' 4 a)'
(2.101)
(2.102)
Prom eqs.(2.10l) we obtain
a, = — -
*,:
c,=
y (1)
(0 V ^2^'
, 12 /fe
rfl = .
(2.103)
(2.104)
(2.105)
(2.106)
Thus, the solution for P can be written in the following form:
1
64\
— ) COS (^ -
y 0)
sintj).
(2.107)
It is easy to see that the quantity — i- is none other than the ratio of
the instantaneous angular velocity n. = kijf to the velocity od. Thus, the coeffi-
cients of the first terms in eq.(2.107) are analogous to those previously ob-
tained for Hy = const, namely
«!= , Oi = —
Y"
The terms containing sin \lr in the expression for P are derived from the /96
influence of inertia forces generated as a result of nominifomi rotation. In
backwajTd rotation, the inertia forces tend to lift the blade which is in the rear
position; this causes a change in the flapping motion and a decrease in angles
of attack so as to attain equilibrium. In so doing the ajd-s of the cone tilts
98
to the left.
So far we have investigated the particular solution for eq.(2.98)» charac-
terizing forced oscillations of the blade. Let us now examine the general solu-
tion of eq.(2.98) without the right-hand member, i.e., the equation
Setting y = 8 in the particular case, we find the solution in the form
p=C,e-*-fC2<l.e-*. (2.108)
The general solution of eq.(2.98) then becomes
X
p = C,e-^ + C,<^e-^ + Y (-y + ^) +
8ki> kii \2k C 2.109 J
-| ~ COS (jj -f- — !- sirn|j sin ijj.
yiii (I) Y**
The values of the coefficients C^ and Cg are fo\ind from the initial condi-
tions ^i = 0; P = 0; p' = 0:
(2.110)
As we see from eq.( 2.108), the terms containing Ci and Cg decay extremely
rapidly; thus, in one revolution (t = 2Tr) the degree of perturbation diminishes
tenfold: e" * = e'^'" « 0.002, te"* « 0.012.
This furnishes a justification to use only the particular solution of
eq.(2.107)» neglecting free oscillations of the blade, a procedure also confirmed
by experiment.
A conparison with experiment showed that, xinder static operating conditions,
the induced velocity distribution over the disk has a substantial influence on
flapping; the refined formiilas for the flapping coefficients are given elsewhere
(Ref.8).
For a rotor with a flapping conpensator, the flapping motion of the rotor
is determined from the formiolas:
/ X . So \ 1 SwM 1 .
1 + -J-Wo, ^"^ 1 + -— tono,
4 4
99
^'^-TO-T'^^'OHi^' (2.113)
c,^-[a,i,...,+±)-T,^l-±t^o,)+J-±,^o,]^^; (2.114)
^.= -[a.(l-i-t.o.) + ^(w.+J-)+^]_J__. (2.115)
A conparison of eqs.(2.1l2) and (2.113) for % and bi followed by conparing
them with the previously obtained expressions (2.103) and (2.104) for a rotor
without flapping conpensator, will show that the lag of the cone and hence of /97
the resultant in a direction opposite to the rotation will be practically the
same, whereas the inclination in a perpendicular direction will decrease.
18. Harmonic Oscillation of the Rotor Axis
Let us now examine the case in which the rotor axis executes a harmonic
oscillation in space at angular velocity
where p = -^ with v being the vibration frequency of the craft.
Since danping of the free oscillations of rotary-wing craft is small, the
harmonic law describes oscillations of the craft close to the true oscillations.
We again obtain the expressions for the mcments of forces relative to the axis
of the flapping hinge. The velocities in the blade section, assuming as usual
M. = 0, B = 1, are equal to
rfp (2.116)
(J = Xisifi -\- Asin pi!^r cos iJj — r
^ dt
The thrust moment is
u
= ^bQa^R^.^(^-[-^-^^+ ^sinp^.cos4> \ (2.117)
2 \3'4 4d(^' 4« /
The moment of the Coriolis force reads
My^= —2 f /ni4sln/7iiu)rsin(jj£/r= — 2/ , (B/\sin/7tjjsirnj;. (2.118)
The moment of the inertia force of rotational acceleration is
100
^ d2v
form
Mjt^= \ in -^^rcos'<^rdr=I^^^ A pm cos p^ cos <^.
The equation of moments, after canceling ty I^.h • o)^, is vo?itten in the
Tr!' + 4'7r + f^= sinp^sinii.+^— sin^cos<l) +
i_A^cos/76cos<b + Y('— +-^V (2.119)
(0 \ 3 4 /
The solution to this equation (assuming that we can neglect the free motion
of the blade, according to the foregoing) is foimd in the form of
B^flo — <^iC0s(/7ilj— -({j) — felsin (pi|j — il») — al cos (ytTtji + ilj) — 6 1 sin (piji + <])).
Substituting this solution into the equation of motion of the blade, we
find the values of the coefficients:
Id
,. A f(/''-2/» + f (P-I )--^^(P-I)
Ol = —
to Y'^
798
' to V^
-rr (P + 1)2 + (P2 + 2p)2
lb
,. yl 8 4 4 2
-^(/'+l)» + (/'2 + 2p)2
Disregarding powers of p greater than the first (since p does not exceed
0.03 - 0.04) and expressing sin (pilf ± if) and cos (pijr ± li ) in terms of the product
of the form sin pi|r cos ■i/ , cos pil( sin \|; , cos p^lf • cos t , sin piji • sin i/ , we
obtain
p==a.4-i- Asln;;tl.cost5.-f— +1 ) — pcos p^-cos'b-^
Y « \ Y •' "
A 4 A
-| sin/^ijisint}!-! /? cos p'ji sin ij).
to Y ^
101
Recalling that
pA cos /?tp
i4sin7^|»=2y,
dQy ^ 1 dQy
dif <a dt
we find that the flapping motion, in the case of harmonic oscillations of the
rotor axis in space, can be represented in the form
^=".+[f^-(^+')
dQy I
dt 0)2
COS li-f-
A^_^±^n (2.120)
[ u ' Y tii "'^ J
Thus, the longitudinal inclination of the cone of the rotor during rotation
in a longitudinal plane vdll be
a.= -^^+(^ + 0^-^. (2.121)
and the lateral inclination becomes
'" 0) Y d^ "^^ (2.122)
These expressions, relative to the magnitude of the terms proportional to
the angular velocity of the rotor shaft, coincide with those previously obtained
for uniform and uniformly accelerated rotation and thus can be used, in the
general case of helicopter motion, for determining forces and moments of the
rotor, danping the helicopter motion.
For a rotor with a flapping con^Densator, eqs. (2.121), (2.122) vary propor-
tional to eqs. (2. 102) - (2.106).
CHARACTERISTICS OF ROTOR AERODYNAMICS DETERMINED /22
BI HINGED BLADE ATTACHMENT
Hinged blade attachment has a substantial effect on the aerodynamics of the
rotor; therefore, an -understanding of the role and pl::ysical meaning of flapping
motion will help the reader toward a better study of the characteristics of
rotor aerodynamics. These questions are presented below. Fxirthermore, a siirple
graphic derivation of formulas for calculating the flapping coefficients is
given.
102
19. Physical Meaning of Blade Flapping
The equation of tlade flapping in steady rectilinear flight can be repre-
sented as
where
?r^; m..= -^rrfr.
"oo ~ 2/h.v, ' ^"-^ J d?
Here, S,,. n is the mass static moment of the blade relative to the flapping
hinge.
As shown above [see eq.(2.33)]» the e^qjression for — ■::- contains the flap-
dr
ping angle p and the angular velocity 7^ ; this demonstrates that the flapping
motion relative to the flapping hinge is danped by aerodynamic forces.
Owing to appreciable aerodynamic danping [with linearization of the equa-
tion, i.e., on the assimption that Cy = acoaj., the average (per revolution) coef-
ficient of J? in eq.(2.34) is equal to —i— B^ « 1 - l.?], the natural oscilla^
tions of the blade rapidly die out and the flapping motion of the blade becomes
a forced oscillation due to the thrust moment. Since the natural frequency of
the blade is close to the rpm of the rotor [the average (per revolution) coeffi-
cient of p in eq.(2.34) is equal to 1], the blade reaches its maximum oscilla-
tion anplitude ipon a variation in thrust moment with the frequency of the rotor
revolutions, i.e., with respect to the first harmonic.
The correlation between the anplitudes of the second harmonic of flapping
and the thrust moment is by approximately a factor of 10 less than for the first
harmonic. Therefore, despite the fact that the anplitude of the second harmonic
of the thrust moment is high, blade flapping occiu?s mainly with respect to the
first harmonic.
Thus, the bulk of the flapping motion of the blade is described by the
equation
EJ = o.(, — a,cos.]j— ftisinij). (2.124)
Let us substitute eq.(2.l24) into the equation of flapping motion (2.123).
The left-hand side of the equation is equal to ao:
^+P=ao. (2.125)
The equation of flapping takes the form
103
m^^^^a^
(2.126)
(p=270''
It follows from eq.(2.l25) that, in each section of the blade, the svrai of
the inertia force of flapping and of the conponent of the centrifugal force /IQQ
normal to the blade axis is proportional to ao and is a constant, remaining un-
changed ipon rotation of the blade al-
though the flapping angle of the blade
changes. This means that the first
harmonic of the moment of inertia forces
relative to the horizontal hinge is
equal to zero. Therefore, as shown by
eq.(2.l26), the thrust moment of the
blade relative to the flapping hinge
should be the same at all azimuths.
Herein lies the basic characteristic of
a rotor with flapping hinges and the
physical meaning of blade flapping with
respect to the first harmonic: The
blade moves about the horizontal hinge
so that, as a resiolt of the redistribu-
tion of aerodynamic forces over the
blade caused by the flapping, the thrust
moment relative to the horizontal hinge
does not change at all azimuths.
Fig. 2. 28 Distribution of Thrust
over the Blade Radius at Identical
Magnitude of Thrust Moment Rela-
tive to the Horizontal Hinge.
20. Redistribution of Aerodynajiiic Forces over the
Rotor Disk due to Flapping
Equality of the magnitude of the thrxist moment of the blade relative to the
flapping hinge at every azimuth will not resiolt in blade thrust, calculated only
with consideration of the first harmonics of flapping which are the same at all
azimuths, since the distribution of thrust over the radius changes from azimuth
to azimuth (Fig. 2. 28). However, owing to flapping of the rotor with hinged
blade retention, the first harmonic of the change of blade thrust decreases
steeply.
The blade thrust depends on the flapping motion mainly with respect to the
additional relative flow normal to the blade axis"" produced during flapping of
the blade elements, which changes the true angle of attack of the element. The
changes which introduce first-harmonic flapping into the distribution of true
angles of attack over the rotor disk are appreciable. For exanple, the addi-
tional vertical velocity of the air AUy = a^r of a blade element at azimuth
i|r = 90° and of the same element at azimuth ;j; = 270° is the same in magnitude but
opposite in direction. However, owing to the difference in the horizontal comr-
ponents of the relative flow, the true angle of attack of the element decreases
■5^ For sinplicity, we will call the velocity of the air normal to the blade axis
the "vertical" velocity.
104
little at \|f = 90° and increases much more at i|t = 270° . This explains the local
increase of the true angles of attack of the blade sections in the region t =
= 270° and the occurrence of flow separation at high flying speeds for a rotor
■with hinged blades (Fig. 2. 29).
Above, we determined the relative vertical velocity of the flow at azimuths
i|r = 90° and i|; = 270°. This was found equal to, respectively, -aircu and aircu.
The ej^jressions have a sinple explanation.
Figure 2.30 shows a rotor whose blades have different flapping angles at
azimuths i|r = 0° and f = 180°, i.e., the axis of the cone of the blades is de-
flected backward (ai ^ O). Here, the blades have a maximum vertical velocity
with respect to absolute magnitude on passing through azimuths ijf = 90° and /lOl
i|( = 270° since the blade, in the same time interval At = ^ , is vertically
cu
displaced by the largest magnitude (p > n) . At azimuths i|r
the vertical velocity of the blades is equal to zero.
0° and i; = 180°,
^'^/SO'
f =90° ■,270°
Fig. 2. 29 Variation in Angle of Attack Fig. 2. 30 Displacement of Blade Section
of the Blade Section with Respect to Relative to Plane of Rotation on
Azimuth, due to Blade Flapping. Blade Turning.
Thus, a change in vertical velocity and, consequently, in true angle of
attack and blade thrust at azimuths ilf and \|; + rr will take place on variations in
TT "^
the blade flapping angles at azimuths i|r + -^ and % + -^ and vice versa. Bear-
ing this in mind, it is easy to understand how the rotor flapping will vary if,
for some reason, a cyclic change of the true angles of attack takes place or an
additional moment relative to the flapping hinges appears on the blades.
For exanple, if because of blade twisting or for some other reason the
angles of attack of the sections increase to a maximum at azimuth \|f and decrease
maximally at azimuth % + tt, then an additional flapping motion of the blades is
105
established so that the blades occupy the lowest position at azimuth i|(
TT
2
and, when flapping upward, reduce the true angles of attack to a value at which
the condition of constancy of thrust moment relative to the horizontal hinge is
observed at all azimuths. The highest position of the blades is at azimuth ^ +
+ -O- after which they drop, restoring the diminished angles of attack. Along
with the variation in flapping with respect to the first harmonic, the forces H
and S also vary (Fig.2.3l):
A// = rA3sin6;
~^S=T^'pcos'!^.
It was shown above that, despite the large first harmonic at velocity U,
the first harmonic of the variation in blade thrust with respect to azimuth is /103
relatively small, since it substantially decreases because of the flapping. The
second harmonic of blade thrust is larger and the third smaller than the first
harmonic .
Plane of
rotat ion
Fig. 2. 31 Variation in Flapping and
Longitudinal and Lateral Forces due
to Dynamic Twist of the Blade with
Respect to the RLrst Harmonic.
P
P"
m
h.h
0.10
-0.10
/
/
^
_
-
/
'"h.h
^
k
--
>
N
\
s.
>
v^
--
\
\
\/
/
1
\
^
/0(N
\
\
200
—
_-
-
\P"
\
\
7102
'300 ^°^
Fig. 2. 32 Variation in Flapping Angle,
Angular Acceleration of Flapping, and
Thrust Moment of Blade Relative to
Flapping Hinge as a Function of Azimuth.
The second harmonic of blade thrust causes second-harmonic flapping motion
of the blade
Afi-= — fljCOs 2()j — ^2 sin St]),
which is equalized by the moment of inertia forces
I^^ »^ (^^ + Ap) = /,., ">= (3^2 cos ^ + 36, sin 2^)
(2.127)
(2.128)
106
\
\
\
^
-
/
/
/
r
■^
-^
[>
-
^
^
^
—
^'•
s 1
N
\
^
20(S
\
300
r
/
\
/
\
/
\
^
^
Fig. 2. 33 Variation in Angular
Velocity of Flapping, Angle of
Attack of Section at r = 0.975,
and Blade Thrust as a Function
of Azimuth.
and creates some redistribution of aero-
dynamic forces of the blade with respect
to azimuth, which is less extensive than
for the first harmonic.
The higher harmonics of flapping
are very small and have practically no
effect on the blade aerodynamics.
The graphs in Figs. 2.32 and 2-33
are an illustration of oior statements
on blade flapping and variation in aero-
dynamic forces with respect to azimuth.
The diagrams were obtained by rough
calculation, on the assutiption of uni-
form induced velocity distribution over
the rotor disk and without consideration
of elastic oscillations of the blade
which affect the magnitude of the upper
harmonics of flapping and blade thrust.
The calculation was performed for
the following initial data:
K=0.30; t^^QAQ; a=-9.4°; Mo = 0.6;
= 0.9; ;fe=0, /,
k.v,"
= 0.
The curves show the kinematic characteristics of flapping P,'— rr-, ^—,
W
dr
thrust, and thrust moment of blade t^, m,,.^ , and angle of attack of section cfp
at r = 0.975.
We see from this exanple and from Table 2.11 that, beginning with the
second harmonic, the flapping coefficients markedly decrease and, beginning with
the third harmonic, the decay coefficients of blade thrust diminish. Thus, the
angle and angular velocity of flapping as well as the angle of attack of the
blade section vary mainly with respect to the first haraionic, i.e., with the
frequency of the rotor revolutions. The second harmonic becomes manifest in
angular acceleration of the blade, whereas the blade thrust and its moment rela-
tive to the flapping hinge vary mainly with respect to the second harmonic .
21. Approxim ate Derivation of Formulas for Flapping Coefficients
On the basis of the properties of blade flapping, described in Subsection 19,
we will derive approximate expressions for determining the flapping coefficients
ai and bi obtained in Subsection 4« For sinplicity, we wiU take B = 1 and /1G4
will disregard small terms of the order of y, so as to obtain expressions with
an accuracy to |j, .
On the basis of the constancy of the thrust moment at all azimuths, we will
equate the thrust moments for azimuths differing by 180° . This method permits
107
a better definition of the mechanism of equalization of thrust moments by means
of flapping, under different conditions of blade flow at azimuths differing by
180°.
The angle of backward tilt of the axis of the rotor cone ai is determined
from an examination of azimuths \lf = 90° and ) = 270°; the angle of sideward tilt
of the axis of the cone toward the side of the advancing blade (\jf = 90°) is de-
termined from azimuths ilf = 0° and 'ij = 1S0°.
The superposition of the translational velocity of flight on the rotary
motion of the rotor is responsible for the different operating conditions of the
blades at azimuths 90° and 270°. At azimuth i|i = 90° the velocities are added
and at azimuth ijr = 270°, subtracted. Therefore, the coefficient a^ is equal to
zero during static operation of the rotor and increases with an increase in fly-
ing speed V (or ti = ^ ^°^ "^
At azimuth ^ = 90°, the relative flow in the plane of rotation is equal to
Ux = (jaR(r + IJ.). Here, in the region of large velocities, the backward displace-
ment of the axis of the rotor cone causes a lifting of the blade and a decrease
in the vertical conponent of the relative flow Uy = u)R(X - air), which reduces
the true angles of attack of the sections.
At azimuth t = 270°, the relative flow in the plane of rotation is small,
while the vertical velocity and the true angles of attack of the sections in-
crease: Ux = cuR(r - |i); Uy = u)R(X + air).
Let us then construct the equations for the elementary thrust moment, take
the integral from r = to r = 1 at both azimuths and, equating the results,
find the expression for ai. We can also equate to zero the moment of the thrust
difference at azimuth \|i = 90° and t = 180°:
where
Hence,
a, = 2ix(Acp-^x). (2.130)
Owing to the velocity difference of the oncoming flow at azimuths j = 90°
and \|r = 270°, the quantity ai will vary even at the same change in angle of at-
tack or vertical velocity for the blade at these azimuths. For exanple, vipon an
increase in angle of attack of the rotor, equal vertical velocities appear at
the blade sections at azimuths 'if = 90° and t = 270°. To have the blade thrust
moment increments at these azimuths identical, the angles of attack of the
108
sections at azimuth \|; = 90° should he decreased further and, at azimuth i|; = 210° ,
increased again. Ob-viously, this viill occur xxpon an increase in a^.
This is an important property of a rotor mth hinged blades: Upon an in-
crease in angle of attack of the helicopter owing to an increase in a^, the
longitudinal force H increases and a destabilizing moment appears relative to /105
the center of gravity of the helicopter, causing an even greater increase in
angle of attack; the helicopter is statically iinstable with respect to the angle
of attack.
AUy^-Vd^
Fig. 2. 34 Variation in Velocity
Uy as a Function of the Coning
Angle ao •
We should note that a^ does not depend
on the inertia characteristics of the
blades, since a^ equalizes the aerodynamic
"asymmetry" in rotor operation.
The presence of the coning angle is
responsible for the difference in vertical
velocity Uy of the air relative to the
blade, at azimuths t = 0° and t = 180°
(Fig. 2. 34).
For a blade in the forward position
(t = 180°), the velocity of the air is directed from the bottom ipward; to reduce
the true angle of attack the blade, on passing the azimuths 90 - 270°, is lifted
tpward. Diur-ing the second half of the revolution the blade drops, which inr-
creases the true angles of attack. Thus, the axis of the rotor cone is dis-
placed laterally, toward the side of the advancing blade (| = 90° )»
Let us now derive the expression for the coefficient bi
ponents of flow around the blade sections are equal to:
The velocity com-
at azimuth i|/ = 0,
at azimuth il; = 180°,
U^^wRr; L/y = u)R {l — bir-\-\).ag).
Equating the thrust moments of the blade at these azimuths, we obtain
*i = ^!^ao-
(2.131)
The coefficient bj equalizes the aerodynamic "asymmetry" caused by the
presence of ag. Since ao depends on the mass characteristic of the blade Yj i't
follows that also bi depends on y .
22. Eff ect of Monuniformity of the Induced Velocity
Field on the Flapping Motion
Next, we will define the variation in the flapping coefficients ai and b^
109
for the case in which an additional vertical velocity appears on the sections
and an additional periodic moment relative to the flapping hinge acts on the
blade :
Af/y=— t/iCOs.]j — f/jSint)'; (2.132)
AM=— Micostj) — TWaSintp. , -_„n
The blade thrust moment is the only moment able to balance the additional
first-harmonic moment caused by a variation in the vertical velocity and in the
moment AM.
The linear thrust of the blade receives an increment owing to a change in
the flapping coefficients by a quantity Aa^ and Abi. In this case, the equation
of flapping has the form
R
^drr = /^^^^i>?ao~M^cos'!^-M^sin'!^. (2.134)
In conformity with eq.( 2.134) » we can examine the following equalities: /106
l\dTr] +^2=(Urr| -M,; (2.135)
lUTr] +M,=r{dTr] -M, (2.136)
or, in dimensionless form,
\o /4.=90° \o Ji
d\^4-7d7\ +--i-^/w,=-LfN7rfA -^S-^. (2.138)
\0 /^=Q° \0 /(ti=180»
The equality (2.137) can also be described differently:
Ceo J l-U'-A=90'' Vrf'-A-270"J Y/v..W'"2 (2.139)
or, expressing in the form of
dr
110
"1 ^- M.\
j Y/w.t,'"^ (2.137)
' 41=270'
dt /dt\ ( dt\ , ( dt\ . ,
ilCf)/''"^ = -^^- (2.141)
The physical meaning of eq.( 2.141) is obvious: The flapping hinge moment,
varying with respect to the first harmonic, is equalized by the moment of the
first harmonic of thrust'"".
Henceforth we will use the equality (2.139) and, accordingly, the
equality (2.1^2):
Lf[(^) -(^-L) l7rff=-2— ^vW.. (2.1^2)
"oo
and determine only the flapping coefficient increments. /107
Let us examine the azimuths i]; = 90° and i|f = 270° (K.g.2.35):
-^Af^) =_(Aa/-fi7,)(r + p.);
-^a(^) -=(Aa,F+t7,)(7-,.);
a^ \dr /^=270°
J-fAf^) ~^(^)
= — 2Aa,r2 — 2f/2r.
We
will give the expressions for [ ^'^ ■] and ( — :^) (with an accuracy to p.^);
\ dv \ ^ dr /i
1 /dt\ - , _
— ( -^ ) =r2 ai — |xX — 2r|X9;
a<» \dr /\
aXdr/i ^ o^ L W/a.v ~ \^A=oJ "" ^''"^^ """ ^'^^ ~
— [ipr2+ r (X +bir — i>-ao)]=riiao — r2bi.
However (-^^) is not equal to (-^r) - (-^) „ since
\ dr /i ^ dr /av ^ dr /|=90
the term
-|j.air in the expression ( — — -j is the coefficient of sin^'i]; and does not per-
V dr ^|=go°
tain to the first harmonic.
Ill
ujS(r*u)
u)R(aa,r*U^)
ujUfr-fi)
ujR(Aa,ri-U^)
Fig. 2. 35 For Deriving the
Expressions for Aai.
1
'J
From the equality (2.139) we obtain
Aa, = — 4 U.r'^dr
.P
Y/h.W<-2
Af,
(2.143)
Next, let us examine the azimuths ijr =
and ilf = 180° (Fig. 2.36):
From the equality (2.142) we obtain
A&i=4 U.r^dr '- ..,,
Using eqs.(2.l43) and (2.144), we then derive the formulas for determining
the flapping coefficient increments, -with consideration of a nonuniform in- /108
duced velocity distribution over the rotor disk.
Vcosoc
u)R(&b,r-u,)
Fig. 2. 36 For Deriving the EjqDressions for Abi ,
In first approximation, the induced velocity distribution can be described
by the equation (see Fig. 2. 6)
v{r, ']()=i) -far CDS']*.
(2.145)
Since the positive direction of the additional vertical velocity AUy is
from the bottom \ip and that of the induced velocity from the top down, a comr-
112
b,
/
16
tg
-=r^^
"^^
/
^y^'''^
J
/
""'"^
T'^ofi
parison of eqs.(2.1A-5) and (2.132) will
yield
,=0. 1
{2.im
0.1
Substituting eq.( 2.146) into eqs.(2.l43)
and (2.144), we obtain the sought expres-
sions:
Fig. 2. 37 Variation in b^ as a
Function of \i,.
Aai = 0;
A^i=4a j rMr = a.
{2.1hl)
Thus, the backward deflection of the axis of the rotor cone vd.ll not change,
whereas the lateral deflection will increase by an angle numerically equal to
the increment of the relative induced velocity at the blade tip in both forward
and rear positions. If we assxmie that a = v"", i.e., that the induced velocity
at the leading end of the disk is equal to zero and at the trailing end equal to
double the mean value, then
Aftj=T) =
ia
4fl2 K(Jt2+ \2 '
while the total value of backward deflection of the cone axis will be
(2.148)
*i = TrH'«o + -
<0
4B2/(
^2+X2
(2.149)
A longitudinal tilt of the induced Arelocity field also affects the magnitude
of the longitudinal and lateral forces of the rotor. Let us derive the formulas
for detennining h and s:
(2.150)
/109
(2.151)
Equations (2.149) - (2.151) can be used at p, > 0.1 - O.O5. Therefore, in
Fig. 2. 37 which gives the curve of bi as a fimction of |j,, the sector from p, =
to iJ, =0.1 contains a broken curve, laid approximately through the points p, = 0,
"- The quantity a, as related to the flight regime, can be determined from data
given elsewhere (Ref .25).
113
at which "b^ = and ij, = 0.1.
By means of eqs.(2.143) and (2.144) we can also obtain the approximate
expressions for determining the flapping coefficient increments during curvi-
linear motion of a heliccpter, which were derived in Subsection 4*
METHOD OF CALCULATING THE AERDDYWAMIC CHARACTERISTICS
OF A ROTOR FOR AZMUTHAL VARIATION OF BLADE PITCH
23* Equivalent Rotor Theory
It will be shown below that a rotor whose blade pitch changes cyclically
with respect to the first harmonic
? = ?o — ViCOSip — 9iSina), (2.152)
can be regarded in the aerodynamic design as a rotor with a constant pitch equal
to cpo, but with a different angle of attack. On this basis, the method of de-
termining the aerodynamic characteristics of a rotor with a pitch variable in
azimuth is called the equivalent rotor theoiy.
The equivalent rotor theory furnishes an explanation for the mode of varia-
tion in rotor characteristics with deflection of_the automatic pitch control
mechanism. The formulas for calculating cpi and cpx in relation to the angle of
deflection of the automatic pitch control and the kinematic characteristics of
the rotor hub are given in Subsections 25 - 28. Data published earlier (Ref .15,
14) were used in presenting the material.
First, let us examine the problem formally:
Substituting eq.( 2.152) into the equation of flapping and, for sinplicity,
retaining only the first harmonics, a series of transformations will yield
Below, in Subsections 23 and 24, only the pitch conponents cpo , cp 1 , and 91
will be contained in the formulas so that, for sinplicity, we will omit the sub-
script "0" of cpo •
From eq.( 2.153), the following formulas are obtained for the flapping /HO
coefficients :
114
ao=Y[Y+T^(^ + l*'>-irH= (2.154)
«i=^ — 7 — (>'+Y'p-^i"^)-^«; (2.155)
i-T<^^
3(1 + -,2)
(2.156)
It is obvious that, on making the substitution X^q = \ - ^itx, eqs.(2.154)
to (2.156) can be rewritten in the fonii
^»=^[^+T^(^+'''^]= (2.157)
' 2 ■
*i = — : -, r '^oP' + ?i
(2.158)
(l+'Y'^') (2.159)
A comparison of these formulas with eqs.(2.40) for a rotor with constant
pitch readily shows that ao and the first terms of the expressions for ai and bi
coincide, provided that both rotors have equal |j. and cp, and that the X of the
rotor with constant pitch is equal to X^q. • Henceforth we will denote all quanti-
ties pertaining to a rotor with constant pitch by the subscript "eq" or "e"
( for e quivalent ) .
The coincidence of the formulas enables us to detemiine the flapping coef-
ficients of a rotor with variable pitch from the_foniiulas for the flapping coef-
ficients of a rotor with constant pitch, adding cpi and cpi :
^o = ^Oe; (2. 160)
«i = «ie-'Pi; (2.161)
ftj = 6,^+9i. (2.I62)
In so doing it is necessary to satisfy the conditions of equivalence of
the rotor with variable pitch and the rotor with constant pitch:
X — ^,[i = X,; (2.164)
9=?e- (2. 165)
115
Now we are convinced that the following relations are satisfied:
_ _ _ (2.166)
(2.167)
or
Acpi/^ = (9iCOS-|--|-», sin.j»)f/^=Af/y.
(2.168)
Actually, on the basis of eqs.(2.l60) - (2.165), we represent both sides /111
of the equality (2.168) in e^^^anded form:
— A'f t/x= (?i cos (jj + (p, sin ij)) (7+ [J. sin •]>);
At/y = AX — r (Afli sin 4) — A*, cos ij>) — (xACo cos t}) -f [xAa, cos2 <J) +
-f l^Afti sin <^ cos ijj=yj[j„— r ( — 9, sin ')' — ?, cos <)>)—
— cpiti. cos2 (j) -f (p,(A sinip cos ijj == (^1 cos ip +"^1 sinip) (r + [X sin <);).
It is obvious that the equality (2. 168) is valid here.
It follows directly from eq.(2.l68) that
Uy Uy^ ^(f Uy
°r=? + -^ = ?e + A?-h— + — '=cp,+-^ = a,^. (2.169)
Thus, the angles of attack at all blade sections for the rotor with variable
pitch and for the rotor with constant pitch equivalent to it are equal.
likewise, we can show that
^^(^\ . (2.170)
dr \d? J^ '
t, = t,^; (2.171)
y_. * (2.172)
t — tj.
Equations (2.168) - (2.170) show that a decrease or increase in linear
thrust, produced by a change in pitch of the blade at a given azimuth, is due to
a decrease or increase in Uy at the same azimuth when calculating on the basis
of the equivalent rotor theory.
At equal thrust coefficients, the relative induced velocities are also equal
ta t.Q
116
-V,, (2.173)
from which it follows, based on eqs.(2.l63) and (2.164), that
X as (AO — v,
X — X,= ti,(a — aj = 9i(j,; (2.174)
«e = « — ?i-
We represent the e^^jression for — ^ in the form [see eq.(3.56)3
dr
1^— ir|+'-,„0;. (2.175)
Using eqs.(2.l68) and (2.170), we find /112
dr \ dr I U, + "^^".J^^ \ dr lu7~U?k ~
~ (-^\ (y, COS 6 + ^1 Sin d>); ( 2 . 176 )
?<i' = 9+j — ^^((picostp + yisin 6);
'"*-^^J'^^J'~3'^^^ = '"*c--^j(?'^°^'!'+^i^'"^)'=''I'J(:f-)'^'^^- (2.177)
Since, for a rotor with flapping hinges, the value of the integral
r f B rjrdr is constant at all azimuths, the integral with respect to i|r must be
equal to zero. Consequently, the average per-revolution magnitude of the torque
coefficients of the rotors is identical:
'"t='«t,. (2.178)
dq
However, at equal azimuths the values of ,_ , qx, and mt^ for both rotors
are not the same and the rotors have a different variable conponent of flapping
motion about the drag hinge.
Let us now derive formulas for deteraiining the coefficients h and s of the
rotor with variable pitch from the corresponding coefficients of the rotor with
constant pitch: h.^^ and s,, . On the basis of eqs. (2.161), (2.I62), (2.171), and
(2.177) we then obtain
AA^= — ^^ /\pcos<li-j-A^4.sin(l)= — ^^ X
X[(<PiC0si}i — (p,sin4i)cos i})-|-(^iCOsi}i+^iSiin)))sin ({)]= —^4, ^j; (2.179)
^e'
117
As^= — ti/ Apsinii — A^i^ cos '^ = t^ <d^.
(2.180)
Consequently,
^h
2r.
-^Z'iv
S = Se + ^e?l-
(2.181)
(2.182)
Finally, let us define the relation of the coefficients of forces in the
velocity axes. It follows from eqs.(2.172), (2.174), and (2.181) that
ty^f.
^^ «i /a + A = /,a^H- /?^ + ^, Aa + A A = ^.,^+ ^, 9, - t^^^ = t^^
(2.183)
(2.184)
Equations (2.183) and (2.184) show that a rotor with different cpi and cpi at
identical p,, X^^, cp has identical ty and t^ . Consequently, at equal p,, ty, t^
the rotor with a cycHc variation of pitch and the rotor with a constant pitch
have equal cp, X^jj, oi^^f but different a. This characteristic of a hinged rotor
manifests itself in that, at equal ty and t^ (at equal f Hying weight, speed, /113
and altitude) but at different ^i (different centering or angles of stabilizer
setting), the helicopter will have different angles of attack and angles of
pitch. This is shown in Fig. 2. 38: The rotor with constant pitch and the rotor
with variable pitch, at equal p,, ty, tx have equal ag(j but different a; conse-
quently, the helicopter with a deflected automatic pitch control mechanism in
the same flying regime will occupy a new position in space.
t,S
Plane of rotation of rotor
with pitch variable
in azimuth
Plane of rotation of rotor with constant
pitch and plane of equivalent
rotor for
aj^(x,=ar^-<p,= a.
Fig. 2. 38 Angles of Pitch of Helicopter at Same Flying Regime
but Different Deflections of Automatic Pitch Control.
An TTppm-tant consequence of eqs.(2.l78), (2.183), and (2.184) is the possi-
bility of mathematically determining the interdependence of the coefficients |j..
118
ty, tx, nit irrespective of whether or not the_rotor has a cyclic variation of
pxtch with respect to azimuth since, for any cpi and cpi, the coefficients ty, t^,
nit do ^'^^ change. This property of the rotor greatly sitiplifies the aerodynamic
design of a helicopter.
The above-derived formulas of the equivalent rotor theory will remain valid
even if they are not derived from eq. (2-153)
and even in the absence of assi:mptions of uni-
formity of the induced velocity field (without
discarding higher harmonics of flapping) and of
other assunptions. Consequently, also here
transformations based on the equivalent rotor
theoiy will hold. The higher harmonics of flap-
ping and the loads acting on the blade in the
thrust plane, for a rotor with pitch varying
as a function of the first harmonic and for a
rotor with constant pitch, are identical if the
conditions of equivalence of the regimes
(2.163) - (2.165) are satisfied.
Fig '2.39 Displacement As of
the Flapping Hinges Relative
to the Plane of the Equiva-
lent Rotor.
The equivalent rotor theory is not appli-
cable in the case of widely spaced flapping
hinges, since relative to the new reference plane, i.e., relative to the plane
of the equivalent rotor, the blades execute an additional displacement As
(Fig. 2. 39) together with the flapping hinges, which does not occur when calcu-
lating a rotor relative to the plane of rotation and is not taken into account
in design formulas .
ip=const
h\
If ''Const
^ . cX=oli(at ipfO)Of
a -(X, y [fi^o
Fig. 2. 40 Reconstruction of the Aerodynamic Characteristics
for a Rotor with Pitch Varying in Azimuth.
Finally, for a rotor with constant pitch all dimensionless characteristics
are defined xsgon prescribing three quantities (jj., \, cp or any other three quanti-
ties), whereas for the rotor with variable pitch five quantities (iJ., \, cp, /114
^i> 9i °r 3-^ other five) must be known for determining the dimensionless
119
characteristics in the related axes.
Thus, it has been proved that the calculation of a rotor mth v-ariable
pitch can be replaced by the sinpler calculation of a rotor with variable pitch,
i.e., stipulating equivalence of the fljdlng regimes ( 2.163) - (2.165), viith sub-
sequent conversion by the above formulas.
The sequence of calculation is as follows:
From the quantities ii, A., cp, 91, cpi which are known for the rotor with
variable pitch, we find p,e»^o>9e*
We then determine a
o.>
■'I.*
mt,.
From the conversion formulas, we find ao, aj,.
The equivalent rotor theory is often used in determining the aerodynamic
characteristics of a rotor from graphs. If the graphs are constructed for a
rotor with constant pitch, their change for a rotor with variable pitch will be
as shown in Fig.2.40. In the graphs for the angle of attack at 9 = const (upper
plots) the curves of t are equidistantly shifted by Aof = cpi, and each point of
the curves of h is shifted by ha = cpx 'to the right and by Ah = -tcpi downward.
On the graphs for rotor pitch, at_Q' = const (lower plots) the marking of the
angles of attack is changed (for cpx f^ 0» each curve corresponds to an angle of
attack greater by cpi), and the curves of h, in addition, are shifted by Ah =
= -tcpi. The graphs of mt, ty, t^, ao, and of higher harmonics of flapping a^,
b„ (n = 2, 3» •••) are modified like the graphs of t, whereas the graphs of s,
ai, bi are modified like the graphs of h. On the graphs of the aerodynamic
characteristics in the velocity axes (the plot on the right in FLg.2.40) for a
rotor with cpx ^ 0, the marking of the angles of attack is also changed.
Plane of
equivalent
7^
Plane of rotation
Fig. 2. 41 For Determining the Position of the
Equivalent Rotor Plane.
Let us now derive formulas correlating the characteristics of the rotor
with variable pitch and its equivalent rotor with constant pitch, on the basis
of geometric relations.
Figure 2.41 gives a side view of the rotor and two blade sections at azi-
120
muths 90 and 270° . If we draw a plane turned about the blade axis through an
angle Z^i|f_9oo = -cpi to the plane of rotation, then the blade pitch relative to
the turned plane will be identical and equal to the mean value of pitch per
revolution. This plane is the plane of the equivalent rotor. The angle of /115
attack of the equivalent rotor a, = a - ^)1. If cp^ ji^ 0, an analogous picture is
obtained on viewing the rotor from the azimuth i|f =0, i.e., the plane of the
equivalent rotor is turned, relative to
-r;- the plane of rotation, through an angle
9i in the side plane of the helicopter.
oTT
Plane of equivalent
^ Plane of rotation
Fig. 2. 42 Velocity and Elementary-
Force Conponents of Blade Sections
in Different Reference Planes.
Thus it is obvious that, for a rotor
with pitch varying cyclically with re-
spect to the first harmonic, we can se-
lect another plane of reference relative
to which the rotor pitch does not change .
Therefore, relative to the new reference
plane we can detemiine forces, moments,
and flapping of the rotor by formulas
derived for the rotor with constant
pitch. In so doing, it must be taken
into account that the new reference
plane has a different angle of attack
and that the results of the calculation
pertain to axes related with it and
should be converted to axes related with
the plane of rotation of the rotor.
This constitutes the geometric meaning of the foraiulas derived above.
The position of the aerodynamic force of the rotor relative to the velocity
vector of flight does not depend on the selection of the reference plane; there-
fore, its conponents on the velocity axes, i.e., lift and propulsive forces, are
equal [see eqs.(2.183) and (2.184)1.
fi 7 ~^=^
Fig. 2. 43 For Determining the Difference Uy - Uy
at Azimuths i; = 0° and ijr = 180° .
Let us now outline the changes occurring when calculating the elementary
forces of the blade section on change-over to the new reference plane.
Figure 2.42 shows the blade section at azimuth i|r . The section has a setting
121
angle cp\|i relative to the plane of rotation and a setting angle 9 relative to /II6
the plahe of the equivalent rotor. The angle of attack of the blade section,
i.e., the angle between the chord of the blade and the vector of the total rela-
tive velocity of flow around the section U, does not depend on selection of the
reference plane [see eq. (2.169)]. The relations between the conponents of U at
small values of kp are equal [see eqs.(2.l66) and (2.168)] rU^^ - U^; Uy^ == Uy +
+ Ux/icp or AUy - -/kpUx .
As indicated above, the last expression shows that any decrease or increase
in load per lonit length of the section due to a change in blade pitch at a given
azimuth for an equivalent rotor with constant pitch is the result of a decrease
or increase in Uy at the same azimuth.
Let us define the reason for the variation in Uy at the characteristic azi-
muths ilf = and 90°. At azimuth i|f = 0, Uy and Uy are equal:
^.r'K-h)-.-'i^)
¥e see from Fig. 2. 43 that, at \|f = 0, the value of p changes on changing to
another reference plane by the same q-uantity as a so that a - P = a^ - 3e • This
r, 77 - r dp /' dp A "
means Uy - Uy^ = - r ^-^ - l^-gf-; J •
If /\q) = J^^ cos t, then the plane of .the equivalent_rotor is inclined
laterally relative to the plane of rotation by an angle cpi , on account of which
Thus, when the pitch of the blade at azimuth ^ = changes by -"cpi^ a change
to the equivalent plane in the calculation will lead to a decrease in Uy owing
to a decrease in the flapping rate relative to the plane of the equivalent rotor
by a quantity equal to cpir.
If /kp = -9i sin i|f, then -||- = (-||-) and AUy = /kp = 0.
At azimuth i/ = 90°, Uy is equal to
If /^ = _9i cos iji, then at ^; = 90° -^ = (-g|-) , a = Qf^ , and AUy = Zkp = 0.
122
If Zkp = -cpi sin i|f, then a - o-, = 91 , -^ - \-^) = -cpi" and /117
iJjy = \>.'fi + r-f, = 9i (f -f p.)= — A?t=90-Z7^^_9o. .
Consequently, when the blade pitch decreases at azimuth cp = 90°, the change-
over to the plane of the equivalent rotor produces the same decrease in aero-
dynamic force as a result of the fact that Uy < Uy owing to a decrease in angle
of attack of the equivalent rotor and an increase in flapping rate relative to
the equivalent rotor.
In conformity with Fig. 2.42, the formulas for converting the load per unit
length in the blade section will be
cir
dt _/ dt \
d'r ~\ d'r )^ '
Thus, all fonnulas of the equivalent rotor theory are in essence only
formulas for converting from one system of axes to another.
24. D erivation of Formulas for a R otor with Flapping; Hinges
as for a Rot or without Hing^. Conditions of
Equivalence of ffi-n g:ed_and Rigid Rotors
In the Glauert-Lock theory, when deriving formulas for the coefficients of
forces, torque, and flapping, the flapping angle of the direction of forces, in
space is reckoned from a plane relative to which the setting angle of the blade
in rotation remains constant. Obviously the plane of the equivalent rotor meets
these requirements.
In this Subsection, we will derive formulas for the coefficients of forces
and torque of a rotor, except that we conceive the hinged rotor as rigid relative
to the axis of the cone described by the blades. In so doing, we will take the
plane of the blade tips as the reference plane rather than the plane of the
equivalent rotor. Relative to this new plane, the blade setting angle changes
in rotation but there is no flapping; this sinplifies the expression for the ve-
locity conponent of the flow past the blade Uy normal to the reference plane.
Since Uy enters the expressions for elementary forces more conplexly than the
setting angle, the formulas for the coefficients of forces and torque in the tip
plane are sinplified.
This method gives individual form-ulas applicable to the calculation of a
-■^ Figxire 2.43 shows the displacements of the blade As relative to the plane of
rotation and plane of equivalent rotor during a half -revolution of the blade; it
is obvious that
f' dpj, ^ dp
\
dt y, dt *
123
rotor both with constant blade setting and mth a setting angle variable relative
to the plane of rotation.
Occasionally, approximate- expressions for the longitudinal and lateral
forces of the rotor enter the aerodynamic calculations and especially the sta-
bility calculations:
h=ta^;
tai; 1
(2.185)
Obviously, these expressions are valid if the forces directed parallel /118
to the plane of the blade tips are
equal to zero, i.e., if the re-
sultant of all aerodynamic forces
is perpendicular to the plane of
the tips.
bloil^
Plane of
'^"^cosj^"'^
rotation y^ //
VsLn(a.*a,)
The obtained expressions for
the coefficients of forces parallel
to the plane of the blade tips are
additions for refining eqs.(2.185).
Fig. 2. 44 Velocity Conponents of Flow
Past the Blade.
Finally, we will derive various
formulas while retaining the as-
sunptions of the Glauert-Lock theory.
Blade flapping can be taken into
account only with an accuracy to
the first harmonic. For a rotor
with infinitely heavy blades (ao =
= t-^ = g = 0, the coefficients of higher hannonics of flapping are also equal
to zero) such formulas were derived by Lock (Ref.37)«
Let us now derive these formulas .
The velocity coirponents of flow past the blade in a plane normal to its
axis (Fig. 2. 44, plane N) are the conponent parallel to the plane of the blade
tips
Ux, . ss (or -|- V cos (a -f ai) sin 6 ^ uj/? (r -j- (i sin <^) = wRUj
"(K)
(K)'
and the component normal to Uj
(k)
i/„
'(K)
\/ sin {a -f Oi) — 1) — V COS (a -f o-i) cos •]> sin ag :
where
= toR (X-f jifli — flojji, cos (]j)=co/?(X^|.j — flgf). cos ■\i) = wRUy
^K)=>-+fliiA;
(2.186)
(2.187)
(2.188)
\f characterizes the velocity of the airflow through the plane of the blade
tips.
124
Let us first examine a rotor
having a pitch constant with re-
spect to azimuth. In the section
normal to the blade ajcis at azi-
muth t, the angle between the
plane of rotation and the plane
of the blade tips is equal to
P
TT
= aj. sin ^ - bx cos i|i
Fig. 2.45 For Deteraiining the Blade Set-
ting Angle Relative to the Plane of the
Blade Tips.
(Fig. 2.45). Therefore, the rela-
tion between the quantities per-
taining to the plane of rotation
and the plane of the tips, in con-
formity with Fig. 2.46, is as
follows :
?(K) = tp — P ^ =<p — a, sin <{) + &, cos <|i; (2.I89)
^-^,K,^^A
^y,K>=*^y + ^-P . =t/, + d/^ (a, sin <}.-*, cos <1>);
4, +.
Tw^T;
Q(K)=Q — T'P „=Q — r(aiSinij.-6,cos<}.).
(2.190)
/119
(2.191)
(2.192)
(2.193)
The expression for the angle of attack of the blade section obviously
should not depend on the selection of the reference plane. We will demonstrate
this, after substituting eqs. (2.189) - (2.191) into the expression for the angle
of attack of the section:
'-'X
Uy <fUx + Uy
'fU. + Uy^^-U^^
+ +
Ux
Uj.
u^
(,-....).,.,..,., ^_^^„^^_^^„_
(K)
f/^
'(K)
U^
'•(K)-
(K)
(2.194)
The expressions for the coefficients of aerodynamic forces of a blade ele-
ment in the plane N have the usual form (see Fig. 2. 46):
("S"), =(SCos*(K) + c^pSin<I?(K,)Z7^«aoo(9(K)t7i + t7„ U^);
125
Ondtting intermediate conputations, we can give the final formulas:
- flopr cos <!> i- flot^^ sin 2<1j i- i^^-^, ,, cos Z^l; ( 2 . 19 5 )
+ (2|iC^p^^r — aoo'f (K)^(K)(i) sinijj + Co (2!i.X(K)ao + 9(K)aoH.r) cos ij> -f-
+ -J- aooao9(K)[i^ sin 2<j; — J- ^^ (c^^ + aoocg) cos 2t, ( 2 . I96 )
where c^p is the average value of c^p over the disk.
To determine -~- we must use eq.(2.193): /120
Jl.=(Jl) +(-^) (a, sin .J,- ft, cos i>). (2.197)
dr \ dr Jw \dr /(k) ^^ ^ ' ^
Substituting cp, with respect to eq.(2.189) into the e^qjression for
(-^i^l and integrating, we obtain
Since, in reality, the rotor blades have flapping hinges, the condition of
flapping motion
I
Y ndrj (2.199)
is satisfied, which yields
2m-
' 3
(V) +T^)=— V- ('+T'^)= ^2.201)
1 + -^.^ 1 2
^=-7^^Vv• (2.202)
i(i + |.^)
Let us derive the expressions for the coefficients of longitudinal and
lateral forces parallel to the plane of the blade tips.
126
The elementary longitudinal and lateral forces are equal to
(-^) =(—^\ sin<j» — ('-^^ On cos 0;
V dr Ak) \ dr JiK) ^ \ dr /(«)
(^) ^-(^) cos4.-(^) aosin^.
\ dr /(K) \ dr /(k) \ dr /(k)
(2.203)
(2.204)
Substituting eqs.(2.195) and (2.196) into these equations and integrating /I2l
the elementary forces over the blade radius and azimuths, we obtain the follow-
ing expressions for the coefficients of average per-revolution forces:
Cxp.^l^ a„
S(K,= - — (ao [y W- -fr (1 +3|x=)] + 3X(K)(ao!^ — ^)} .
(2.205)
(2.206)
With consideration of these expressions, the coefficients of longitudinal
and lateral forces of the rotor
are equal to
(2.207)
(2.208)
Fig. 2. 46 Velocity and Elementary Force
Conponents of the Blade Section for
Different Reference Planes.
Calculations show that use
of the approximate formulas
( 2.185) leads to an error in de-
termining h, equal to 10 - 30^
(toward the side of a decrease,
a larger figure always pertains
to small thrust coefficients t) .
The derivatives of the coeffi-
cient h with respect to |jb, a, cp
can be determined from eq.(2.185).
Determination of the coeffi-
cient s and its derivatives by means of eq.( 2.186) gives & resiilt differing
greatly from calculations by means of eq.(2.208). Equation (2.208), just as
eq.(2.6l) of the Glauert-Lock theory, at -uniform induced velocity distribution,
only approximately determines s, but calculations by means of eq.(2.208) are
closer to the experimental data than calculations by means of eq.(2.185).
Practical calculations show that the value of the coefficient s in autoro-
tation is close to zero and amounts to only a small portion of the value of the
product tbi . At average values of m^ (horizontal flying regime) the value of s
is smaller than that of tbi and we can roughly consider s = ^ tbi . At maximum
power conditions, the value of s is equal to or higher than tb^ .
The torque coefficient of the rotor is determined from the expression
127
(2.209)
Since eq.(2.209) provides for integration -with respect to ilf vdthin limits
from zero to 2tt, all harmonics of the expression for ( — —-) vanish on inte-
gration. Therefore, it siiffices to substitute into eq. (2.209) only the center
portion of i — —] :
(K)
•' .2 2 ,2 / 1 1 /, \;1 (2.210)
Substituting eq. (2.210) into eq.(2.209) and integrating, we obtain
We will demonstrate that mt = m^ . In fact, using eq.( 2.197), we find ra.^:
viC )
... , /i22
^ '"* (K) "'"^ J ^"' ^'" * ~ *' "^^^ *) ^'l' J "S' '^^'^'
(2.212)
It is known from the equation of blade flapping relative to the flapping
hinge that the expression for the thrust moment of the blade f J —tzt rdr^ does
not contain first harmonics ijr ; consequently, the integral with respect to ilf is
equal to zero. Thus, the coefficient of the average per-revolution torque m^ is
determined by eq.(2.2ll) (the instantaneous values of m^, are not mutually equal,
i.e., m* 5^ mt ).
It is also easy to prove the equality of the ejqDressions for mt [eqs.(2.2ll)
and (2.48)] 'by taking their difference:
128
The expressions in the brackets are equal to zero, since they represent
formulas [eqs.(2.201) and (2.202)] for determining the flapping coefficients aa
and bi# Thus, it has been proved that m,. = m^
( k)
The lift and propulsive forces of the rotor do not depend on the mode of
calculating the conponents of. the resultant force of the rotor in the related
axes, whether relative to the plane of rotation or relative to the plane of the
blade tips. They are equal to (considering cos ai ~ 1, sin ai « ai;
^y =/(K)Cos(a-^-ai) — A(„)Sin(a4-ai) = ^ cosa — Asin a = ^ • (2.213)
(K) *
^A-(K) = ^(K)Sin(a + a,) + A(K)COS(a + a,) = ^sina + Acosa = ^^. (2-214)
The formulas derived in this Subsection are sinpler than those of the
Glauert-Lock theory. They are also of interest in that they permit tracing the
manner in which, and the factors by which, the formulas for calculating a rigid
rotor (sinplest case) are transformed into foiroulas for a hinged rotor. The
change in formulas takes place for the following reasons:
1. Change in angle of attack of the rotor owing to deflection of the angle
of rotation of a rigid rotor through an angle ai . In place of \ we introduce
X, . = X + aiiJ, into the formulas for a rigid rotor:
^(K) = aoi
-+■
^(K) I V / i , 3
(■H^f
A(K)= — ^ H'^(K)?;
S(K) = 0;
2. Cyclic change of rotor pitch relative to the deflected plane of rota- /123
tion of the rigid rotor:
?(K)=<P-9,(K)C0S'1' -?i(K)Sin<l'.
With consideration of the cyclic change of pitch, the fonnulas for a rigid
rotor take the form
'-='•- [V-¥+f(>+T''f
«(K)= — - h'<)\?\>- ^/+'^ — '
S(K)=— — >-(K)?l(K);
129
m
,,= -T^-(^-+T'^-N^)+^(i + ^^).
where cpi = ^it ^>i - -^i ^o^ "the case in which the blade pitch does not
change relative to the axis of rotation.
We note that, in calculating a rigid rotor with variable pitch, it is im-
possible to use a new reference plane relative to which the pitch is constant,
i.e., the equivalent rotor theory. This is due to the fact that the blades, on
rotating, do not lie in a new reference plane but actually leave it, i.e., per-
form flapping motion relative to it, and the rigid rotor theory does not hold
for the new reference plane.
3. The coning angle of a rigid rotor in forward flight creates a cyclic
change in the velocity con^jonent Uy of the section flow (see Fig. 2. 34) J AUy =
= -|i.ao cos t. Furthermore, owing to the presence of the coning angle the blade
thrust is projected onto the plane of the tips, stpplementing the forces h^^^
and S/^x . These factors further conplicate the formulas, so that they acquire a
form which, after the substitutions cpi = -b^ and cpid^^ = ^i, coincides with
that of eqs.(2.198), (2.205), (2.206), and (2.211):
^(K) =
"""[t
) "fl(K)l^
H
i+fi^^
2 '
fw--'-^(l+3(.^)
'"t(K) =
"" (1 A I 2 ?i(K)M'\ , aoii ,,
x(aot^ + -|-?.,K,)]+^(l+l^^).
(2.215)
(2.216)
(2.217)
(2.218)
4. Change-over from the plane of the blade tips to the plane of rotation /124
according to the expressions
h=--h^K)-'!-ia^;
ntx
tn
t(K).
(2.219)
)
130
When using eqs.(2.2l5) - (2.218) for calculating a rotor with flapping
hinges according to the condition of flapping motion [eq.(2.199)], ^o, ^ir^) >
and 9l(l^^ will be equal to
ao=Y
\k) . '?l(K)f^
f^(l + !^=)
'i'lCK)^-
4go|j.
2 '^
2f^
1 u2
2 '^
x(^('<)-'-?i(K)l^+y ?)•
X
(2.220)
Let us next examine a rotor with pitch varying in azimuth.
Equations (2.215) - (2.218) without any changes are applicable also to
calculating a rotor with variable pitch.
In this case, the conditions of equality of the setting angles relative to
the plane of the blade tips are
(2.221)
where 9l(^^ > 9i(ic) f ^^^ ^o* a^s before, are detemdned in conformity with the
condition of zero moment at the flapping hinge in accordance with eqs.(2.220).
In place of the term A.^ ^. - cpl(JJ^^J' it is convenient to substitute A. - cpiij, into
eq.(2.220).
Thus, the calculation of a rotor with variable pitch is acconplished by
means of formiilas derived in this Subsection and differing only by the fact that
ai and bi are not equal to cpi^j^^ and Ti(i(.) ^"d are found from eq.(2.22l) after
determining cp^, . and cpl(^^ •
The angles 9i(ic) and 9i(^n should be equal to the angles between the plane
of the blade tips and the plane relative to which the rotor pitch is constant,
i.e., the plane of the equivalent rotor. Consequently,
<?•(«)= -\.
= a,
(2.222)
Thus, without introducing the concept of an equivalent rotor we obtained /125
131
eqs. (2.220) and (2.221), after actually relating the quantities pertaining to
the plane of the blade tips vjith their corresponding quantities of an equivalent
rotor.
The formulas derived in this Subsection yield the conditions of equivalence
of a rotor vdth and -without flapping hinges: Rotors are "equivalent" if their
angles of attack differ by a quantity equal to ai, and a rotor without flapping
hinges has a coning angle and conponents of cyclic pitch change determinable by
eq.(2.220) or eq.(2.22l).
Here, it is assumed that the flapping hinges are located on the axis of
rotation of the rotor or close to it and we can disregard the effect of second
and higher harmonics of flapping on the aerodynamic characteristics of the rotor.
The geometric meaning of the conditions of rotor equivalence is that, ipon
satisfying these conditions, the position of the blades of both rotors relative
to the velocity vector of the oncoming flow and their setting angles at all azi-
muths are identical. It is obvious that, in this case, the thrust moment of the
blade relative to the axis of rotation of a rotor without flapping hinges is
equal to zero.
If, for a helicopter with a rotor without flapping hinges, the cyclic varia-
tion of rotor pitch for balancing the longitudinal and transverse moments is
such that eqs. (2. 220) are not satisfied, then the aerodynamic characteristics
of the rotor differ from those of a rotor with flapping hinges. For exanple,
by creating a transverse moment by a lateral shift of the center of gravity of
the helicopter toward the side of the advancing blade (ij/ = 90°), we can reduce
the angles of attack of the blade sections at azimuth \|f = 270° and thus eliminate
flow separations for a rotor without flapping hinges.
25. General Expressions, for Determining the Components
of Blade Pitch Change cpo> 9i > ^^'^ ^1
In Subsections 23 and 24, we presented a method of calculating the aero-
dynamic characteristics of a rotor with a blade pitch cyclically varying in the
first harmonic
?=9o — ?iCostJ- — tpiSin<|.. (2.223)
Let us now derive fonnulas for determining the cciiiponents of blade pitch
change cpo> 9x > ^^^ cp^ .
The blade pitch established by the control units of the helicopter - control
of the overall rotor pitch and inclination of the automatic pitch control mecha-
nism - is represented in the form
e = 9n-&iSirnJ>-62COS>p. (2.224)
We assume that the design and working principle of the automatic pitch
control are known to the reader [see, for example (Ref.l2)].
132
In addition, the blade pitch of helicqaters is iisually changed during blade
flapping, which is achieved by a special arrangement of blade tiorning levers and
flapping hinges. The hubs of such rotors are called hubs with "flapping com:-
pensator". Let us examine several schemes of hubs with a flapping conpensator:
Cardanic and non-Cardanic hubs differing in control of blade rotation about the
axial hinge, and also hubs with an offset and with a turned flapping hinge
(Fig. 2.47).
a.r
Co ar
Pm^P
/126
a.r
Fig. 2. 47 Schematic Sketches of Rotor Hubs.
a - Cardanic; b - Nonr- Cardanic ; c - With offset
hinge; d - With turned hinge; v.h = Vertical
hinge; a.r = Axis of rotation;
h.h = Horizontal hinge.
In the first scheme (a), the blade turning lever does not participate /127
in moving the blade relative to the drag hinge but participates in others. In
the thii^i scheme (c), the flapping hinge is located such that, in horizontal
flying regimes, the blade axis is practicaUy perpendicular to the axis of the
flapping hinge and goes through the middle between its bearings.
In these schemes, the interdependence of setting angle and flapping angle
of the blade is acconplished by displacement of the ball bearing of the blade
lever A from the axis of the flapping hinge (n / O). In the fourth scheme of
the hub (d), the interdependence of pitch and flapping angle is achieved by
rotation of the axis of the flapping hinge.
In all schemes, the blade is shown in a position inclined about the drag
hinge through an angle Cq = lav • The flapping angle of the blade p is in a
plane perpendiciilar to the plane of rotation and goes through the axis of the
blade. Since the angle Cq is small, the angle of turn about the flapping hinge
in the first three schemes can be considered equal to the flapping angle of the
133
blade: P^.h =
P
cos Co
- p and, in the fourth scheme, as equal to
K..--
cos (c, — Co)
(2.225)
We derive the forrmulas in the follovdng sequence: First we determine the
mode of blade pitch change if the axial hinge had seized and the ball bearing
of the blade-turning lever A was discon-
nected from the rod of the automatic pitch
control mechanism. This change in blade
pitch, taking place without turning of the
blade in the axial hinge, is called "kine-
matic change of pitch" . We will denote it
W /^ i n •
We then determined the amount by which
the blade is tiorned in the axial hinge,
owing to the fact that the point A is con-
nected by a rod with the automatic pitch
control and cannot be displaced in flapping.
This change of pitch is designated by /kpa.h .
The overall change of pitch Acpo is equal
to the sum of /kpi^jn and Zkipa
Acp^ = Acpki„ + At?
a.h
(2.226)
Fig. 2. 48
Kinematic Change of
Blade Pitch.
The kinematic change of blade pitch in
flapping is due to the blade axis being per-
pendicular to the axis of the flapping hinge.
Its derivation is clear from Fig. 2. 48.
Point B, referring to the leading edge of
the blade, during flapping of the blade is displaced relative to the plane of
rotation by a greater amount than point B' referring to the trailing edge. Conr-
sequently, the blade changes its angle relative to the plane of rotation:
A?kin=-T-
As
b
_ _i_sincolhA- —
PtonCQ.
(2.227)
The derivation of /kpkin is illustrated also by the drawing shewn in
Fig. 2.49.
It is obvious that, when the flapping hinge rotates together with the blade
during flapping of the blade relative to the drag hinge (hubs of Sikorsky heli-
copters. Fig. 2. 50), there is no kinematic change of pitch. Such a change is
virtually absent in the scheme shown in Fig. 2. 47c (hubs of Mil' helicopters)
since, in horizontal flying regimes c
hut
A9|^.^ ^^Ptan(Co-
Co;
-O-O-
(2.228)
In the scheme of a hub with a tijirned flapping hinge (scheme d in Fig. 2. 47;
such a scheme for the drag hinge is sometimes used for tail rotors of single- /128
rotor helicopters),
134
A?ui,s=~PM<'i-Co).
(2.229)
Wow let us find Acpa.h • ^ point A in Fig. 2. 47 were not connected by a rod
with the automatic pitch control mechanism, then during flapping of the blade
it would be displaced relative to the axis of rotation by an amount As = nP for
a Cardanic (universal) hub and As = [n +(lv.h - n)c| + I^CqIP » [n + t^,co]3 for
Fig. 2-49 Kinematic Change of
Blade Pitch.
cw
Fig. 2. 50 Schematic Sketch of Hub of
Sikorsky Helicopter.
a non-Cardanic (nonuniversal) hub. Since point A cannot have such a displace-
ment, the blade in flapping turns about the axial hinge by an amount of
As n a X a
(2.230)
(2.231)
(2.232)
for the universal hub and
A9^^ = -(-p + '^o) P= -(tanai + Co)P
for the nonuniversal hub.
For a hub with an offset flapping hinge, we have
A?a.V, -= - [tan =>! + (Cq - C^J] P « —tan OiP.
For a hub with a turned hinge, Apa.h = 0.
Thus, the total change of blade pitch during flapping motion is equal to:
for a universal hub
Acp^ = Acp^.„ + Atp^.v, = ( — txn o, +t»n Cq) P«s — (w o, — Cq) p; ( 2 . 233 )
for a nonuniversal hub and a hub with offset hinge
A<p,==-tanai.p; (2.234)
for a hub with turned hinge'""
-" For sinpllcity in Subsection 16, we take Cq = 0, Acpg = -P tan g^
135
A9c=-tan(oi-Co)p«-(ta„Oi-Co)p. (2.235)
The dependence of pitch on the angle P, in the general form, is esqjressed by
A'ft=-*?. (2.236)
where k is a coefficient of the flapping conpensator.
The value of k is determined from eqs.(2.233) to (2.235).
Oiir derived expressions for k do not take into account additional changes
in the setting angle, such as those caused by inclination of the rod of the
automatic pitch control mechanism, etc. Therefore, the quantity k should be
corrected by measurements on a manufactiired hub or its model. This is deter- /129
mined as the partial derivative k = _2x_ at an average blade pitch and blade
angle of deflection relative to the drag hinge equal to Cq.
For further confutations, the quantity k is conveniently represented as the
tangent of some angle 6 :
^(p(With consideration
of elast ic twi st)
Average elastic twist
Vq per revolution
Deformed
blade
Unde formed
bl ade
Ai=tftn8.
(2.237)
/a r'
Fig. 2. 51 Rotor Pitch and Blade
Twist, vdth Consideration of
Average Elastic Twist per
Revolution.
The blade pitch is equal to the sxmi
of the angle 9 established by the conr-
trols and the angle bss} c '
9=00 — OiSiniJ. — ejcostp— ;fep. (2.238)
Substituting into eq.( 2.238) the
expression for P with an accuracy to the
first harmonic, we obtain
^ — (e„ — Aao) — (6 , — A6j) sin (jj —
— (Oj — Afflj) cos ij).
(2.239)
Conparing this with eq.(2.223), we find
(2.240)
We recall that, in the calculations by the Glauert-Lock theory, the blade
pitch is coiinted from the zero- lift angle of the profile:
P^l.r.t-
■«0 = 6n — ^^n — a„
(2.2400
136
For a rotor with a flapping conpensator, the equivalent rotor theoiy does
not take into accoimt the change of blade pitch with respect to harmonics higher
than the first, in view of the fact that this change is produced by higher har-
monics of blade flapping. Higher harmonics of the change of pitch can be ac-
counted for by specially derived forroulas.
This also pertains to the average elastic twist over the blade with respect
to higher harmonics and also to higher haraionics of the change of pitch due to
elasticity of the automatic pitch control mechanism. The average elastic twist
over the blade in the first harmonic Vi and Vn must be introduced into the
* a V a V
expressions for cpi and cpi . The average twist per revolution vq must be subdi-
vided into average twist over the blade Vq and variable twist over the blade
radius Vq - Vq ; the first is introduced into the expression for cpo, and the
second is added to the geometric twist of the blade Acp (Fig. 2. 51) •
Averaging of the elastic twist of the blade (this can be determined in
flight tests or by calciilation; an estimate of the magnitude of twist can be
made from the magnitude of the hinge moment of the blade) is carried out by
means of the formulas
/r^
j vor2dr
_o
1
1
K =3i•V^^^^«^>;=o.r•
(2.2A.l)
form
With consideration of the elastic twist of the blade, eqs.(2-240) take the
(2.242)
(2.243)
(2.24^)
With the use of eqs.( 2.161) and (2.162), we find
&v
Solving this system relative to cpx and cpi we obtain
?r
1+A;2
l+ft2
1 +*2
(2.245)
137
1+^2 1+^2 1 + ^2 • (2^246)
26. Determination of Flapping _Coef ficients of
Rotor with Flapping; Compensator
After substituting eqs.(2.245) and (2.246) into eqs.(2.l6l) and (2.I62), we
obtain the following e^qjressions for the flapping coefficients ax and bi :
^ _ ai, + ^»i -61+^6; , -^'av+^ "^'av . (2.247)
' 1 + A2 "T" 1+^2 '^ 1 + *2
^ »,,-fe^, e, + A6i , %v+^^iav (2.248)
1+^2 ' 1 + /fe2 ' l+k2
The first addends on the right-hand side of eqs.(2.247) and (2.248) deter-
mine the flapping motions of a rotor having a flapping confjensator td-th an unde-
flected automatic pitch control mechanism G2 =61 =0 and without consideration
of elastic twist. For sinplification of the formulas we write them in the form
- _ aie + fe^ie . (2.249)
U-i — ■ ,
' 1 + ft2 '
*'=TT^- (2.250)
The presence in the kinematic scheme of the hub of a flapping conpensator /I3I
greatly affects the flapping coefficients of the rotor. Upon an increase of k
the inclination of the axis of the rotor cone to the side of the advancing blade
_ bi -
(i|f = 90°) bi decreases. When k = ° we have b^ = 0, and with a fiorther in-
1.
crease of k, bi becomes negative, i.e.- the axis of the cone is incHned to the
side of the retreating blade (^ = 270°).
The backward inclination of the axis of the rotor cone aii ipon an increase
in k varies differently. At a small value of the coefficient of the flapping
conpensator (k < -r-^ — ), the axis of the rotor cone is still inclined toward the
advancing blade and the setting angle of the blade increases at azimuth i|f = 90°
(A9k = -kp) and decreases at azimuth \|f = 270°. Therefore, the coefficient ai
increases. At k = —-ia— , bi = and the setting angle at azimuth t = 90° and
t = 270° will not change. Thus, ai = ai . On further increase in k, the angle
b^ becomes negative and the setting angle at azimuth i|f = 90° decreases, whereas
at azimuth i|f = 270° it increases. Therefore, the coefficient aii decreases.
138
For a single-rotor helicopter, the tail rotor has no automatic pitch con-
trol so that the blade flapping is determined by the quantities ag, a^, hi (the
elastic twist of the tail rotors being small) .
The maxom-um flapping angle of the blade in this case is equal to
P.na. = «o+AP=«o + l^afT^5 = ao+^'^!^^ (2-251)
Equations (2.249) - (2.251) show that the flapping conpensator decreases
the magnitude of the variable portion of the flapping motiori^^ Ap and changes the
azimuth at which the flapping angle has an extreme value tan \|rR = _-^ . Both
m 1 n ai
the first and second factor may be of significance for a single-rotor heUcqpter:
the first decreases the variable loads on the blade of the tail rotor and the
second changes the gap between the blades of the tail rotor and the tail boom.
For helicopters of side-by-side configuration, the gap between the blades
and the fuselage is deteraiined mainly by the quantity bi . It is expedient to
select a kinematic scheme of the automatic pitch control mechanism such that a
deflection of this mechanism will not influence the quantity bi (usually only
course control of a helicopter is acconplished by the automatic pitch control,
i.e., change in ai). Consequently, for helicopters of side-by-side configura-
tion we must take into account, when selecting the magnitude of the flapping
conpensator and disregarding the elasticity of the automatic pitch control and
of the blades bi = bi, that the quantity b^ should be higher or lower depending
on the direction of rotation of the rotors.
For coaxial helicopters the 'gap between the blades of the ipper and lower
rotors depends on the quantity b^ (because of the mutual interference of the
rotors, the operating conditions of the ijpper and lower rotors are not the same
so that also the difference of the coefficients ai of the rotors has an influ-
ence on the gap). As we see from Fig. 2. 52 this gap is
A = // + /?P.f. -/?Pu-.=^-^(^.y. +*>,...>•
It is obvious that to increase the gap we must select a magnitude of the /132
coefficient of the flapping conpensator such that b^^ ~ ^ii =0.
The value of b^ of single-rotor and fore-and-aft helicopters and the value
of a^ of helicopters of any configuration are deteraiined by balancing the heli-
copter. The pilot, deflecting the control stick and acting on ai and bi ,
^;- R.A.Mikheyev determined that Ag decreases somewhat less than Vl + k^-fold,
since tail, rotors with k = and k ^ should be considered at an identical
angle of attack and since of, of a rotor with k ^ is less than a^ of a rotor
with k = 0; due to this fact, the quantity ai^ of the former, at equal thrust,
is greater than that of the latter.
139
'pp er rotor
Og-bt
Fig. 2. 52 For Determining
the Gap between Rotors of
Coaxial Helicopters.
establishes ai, bi, h, and s in such a manner
that the helicopter is in balance. However, at
the required values of ai or bi the quantities
ai and bi depend on aii and bi ; consequent-
oon oon ^
ly, the flapping conpensator influences the de-
flection of the helicopter controls in flight,
i.e., its "balancing curves".
27* Determination of the Components of Blade
Pitch Change cpi and ^i after Deflection of
the Automatic Pitch Control
The second addends on the right-hand side
of eqs.( 2.245) and (2.246) determine the incre-
ment of the conponents of cyclic pitch change and the flapping coefficients of
a rotor with a flapping conpensator, after manipulation of the automatic pitch
control. They represent the change in position of the blade cone and the direc-
tions of forces and moments relative to the rotor shaft when the helicopter
controls are manipulated.
¥e denote these by cpi and cpi
" "'• oon "■'•CO n
or a^
and bi
^' con ' con
— Bi + yfeBa
1 +ft2
= *, =
6.; + feO]
"" 1+ ft2 •
(2.252)
(2.253)
Let us now establish the relation between the angles of inclination of the
automatic pitch control and the magnitude of the angles 9i and Gs .
Figure 2.53 shows a diagram of the hub and automatic pitch control in top
view along the rotor shaft. The arrangement of the flapping hinges is not shown,
since this has no influence on our derivations (only the expression for k de-
pends on it) . The segments AA.' are projections of the inclined rods of the
automatic- pitch controls; point A is the coipling of the rod with the blade
turning lever, while point A' is the coi:pling of the rod with the automatic
pitch control itself.
The rotor blades are shown in positions at which Acpeon = ^s (if = 180°) and
Acfbon = 01 ('I' = 270°); here the hub and automatic pitch control are turned, rela-
tive to the longitudinal axis of the helicopter, through the angles ^^^^ =
= 180° + Co and %^^y, = 270° + Co-
The angles of turn of the automatic pitch control will de denoted by the
letters k and Tj, where k is the tiirn mainly causing deflection of the blade cone
in the longitudinal plane of the helicopter and Tj in the transverse plane.
Let the automatic pitch control be deflected through an angle k relative to
the axis OO' located at an angle Ailfoon "to the transverse plane of the helicopter.
140
ili=t80
AfBi
if)=270'>
\ Longitudinal axis
\0 t of helicopter
Fig. 2. 53 Kinematic Diagram of Hub and Automatic
Pitch Control.
7133
Considering that the vertical displacements of the points A and A' are
equal to (S,^ = Sj^), we find
6^^> = ^=*^?5:f<
02 = — =*-pfsin (a.i-Atcon -Co)=Z>2x;
s!")
.R
= -x--^COs(a„-A<l'„n -Co) = -£'i'<-
(2.254)
Similar expressions are obtained on deflection of the automatic pitch con-
trol relative to the axis 00" through an angle T]
(2.255)
Let us determine the increments of the flapping coefficients of the rotor
when the automatic pitch control is deflected through the angles k and T), sub-
stituting the obtained relations into eqs.(2.252) and (2.253;
,,, ^ /?a.p cos(oii — A^/^^^ — co) + fesin(aii-Atc,n — gp) _
l+*2
«a.p
a\
(1)
=x ^ cos 8 cos (0,1 - A-P^n - Co - 8);
»b
- Ti ^ cos 8 sin (0,1 - .^ - Co - 8);
* '?.n= * y* ^°^ ^ ^^" ("" -' 1*000 - '^o - 8);
(2.256)
(2.257)
(2.258)
141
•c»n ' l^ ^ " -l-con (2.259)
Usually, the angles are so selected that cos (an - Ailfcon - co - S) >
> sin (ctxi - Moon ~ Co - S)» consequently, the deflection of the automatic
pitch control through an angle h mainly causes a change in the coefficient a^, /134
whereas a deflection through an angle T] will change the coefficient hi . The
product I ^^ cos 6 cos (on - Moon - Co - S)» which is dependent on the kine-
matic scheme of the hub, constitutes a relation between the longitudinal in-
clination of the axis of the rotor cone and the angle h . This is denoted by D^ :
D^=^coshcos{o^^~^^^,„-Co-^). (2.260)
The product ^ ^^ ' cos 6 sin (cth - Ai|/con - ^o ~ ^) characterizes the in-
clination of the axis of the rotor cone in a lateral direction. TMs is denoted
by Dg:
Dj^^:Pros«sin(a„-Atc,„-^o-8)- (2.261)
'b
The value of the coefficients D^ and D^ can be refined by testing the full-
scale hub or its model. For this, the blades are set in an azimuthal position
(shown in Fig. 2. 53) and relative to the flapping hinge at an angle P = Pav = ^-o.
After deflecting the automatic pitch control through an angle k, the increments
of the setting angles, i.e., the angles Bg and 9i , are measured.
The values of D^ , Dg, D^ , and Dg are found from the e:!^ressions
X
' 1 + A2 X 1 + ft2
^ ^ Dj — kPi ^ 1 62 + Ml
^ l+k2 X l+k2 '
The quantity k is also found from tests (see Subsect.25).
Thus,
?icoa=-^'co„ = -'^i''+^2Ti; (2.262)
^'con = \.n='^l^ + ^2-''- (2.263)
142
Helicopter designers often acconplish the kinematics of the automatic pitch
control in which D^ = 0. This is done so that, with a longitudinal deflection
of the control stick causing inclination of the automatic pitch control by an
angle k, only the coefficient ai is changed, i.e., so that the resultant force
is deflected strictly in the longitudinal plane of the helicopter. This creates
a moment relative to the center of gravity of the helicopter, also acting in the
longitudinal plane of the craft. However, the motion of helicopters is so in-
terconnected in all directions that there is no sense to rigorously insist on
the condition of coincidence of the directions of action of the moment and de-
flection of the control stick.
For helicopters of side-lDy-side and coaxial configurations, for which
special demands are made on the quantity bi, the coefficient Dg should he equal
to zero so that hi does not change when the automatic pitch control is deflected
longitudinally foCTsrard or backward.
We see from eq. (2-261) that Dg = 0, when
(2.264)
If, in the kinematic scheme of the hub and automatic pitch control, a value
of axi - Co - S 7^ is obtained, then the
plane of inclination of the cone axis will
not coincide with the plane of inclination
of the automatic pitch control but will lead
the plane of inclination of the automatic
pitch control by an angle of
Longitudinal axi
!:=c
-c„ — 8.
(2.265)
Fig. 2. 54 Position of Blade at
Instant of Maximimi Pitch Change
(Hub without Flapping Gompenr-
sator) .
Let us explain the derivation of the
lead angle, assimiing - for sinplicity - that
the coefficient of the flapping conpensator
is equal to zero (6 = O). In this case, the
cyclic change of the setting angle is pro-
duced exclusively by inclination of the auto-
matic pitch control.
Figure 2.54 shows a blade in a position
at which its setting angle has a maximum
value since the point A' of the rod, connect-
ing the blade turning lever with the auto-
matic pitch control, lies in the plane of
inclination of the automatic pitch control. The plane of inclination of the
cone axis is perpendicular to the blade position. We see from Fig. 2. 54 that
C, = a-ii - Co . The angle C, is nonzero since the mechanism changing the pitch is
designed such that, for a maximum change of blade pitch at some azimuth t> the
automatic pitch control will be deflected at an azimuth differing by an angle
90° - an + Co . If CTii 7^ Co, then the plane of inclination of the automatic
pitch control will not coincide with that of the cone axis .
At 6 ^ 0, the cyclic change of the setting angle is not only directly due
143
to deflection of the automatic pitch control but also to the fact that the
change of flapping, caused by deflection of the mechanism, in turn changes the
angle of blade setting. Here, the azimuth of the maximum total change of setting
angle lags by angle 6 = tan~''"k behind the azimuth of the maximum change of set-
ting angle due directly to inclination of the automatic pitch control. There-
fore, at 6 f^ the lead angle is determined by eq.(2.265;.
For the plane of inclination of the cone axis to coincide with the longi-
tudinal or with the transverse plane of the helicopter (so-called "independent"
control), the axes of inclination of the automatic pitch control 00' and 00"
should be turned to the longitudinal and transverse planes of the helicopter
through an angle M/aon = £♦
It follows from eq.(2.26o) that, with "independent" control, the coeffi-
cient will be
^'=lf''''- (2.266)
b
If for a hub cq = the rod of the automatic pitch control is vertical, then
Z126
^=—1- (2.267)
'b cos an
and
n cos 8 / . N
^'=7^- (2.268)
28. Sequence of Aerodynamic Calculation of .a Rotor
with Variable Pitch
Thus, the eDspressions for the conponents of cyclic pitch change are written
in the form
— — — - _ v"i +ftv, (2.269)
Vi^-DiX-j-D^n-kbi +v, =-DiX+D2Ti-Afti
l+*2 (2.270)
For brevity, the last addends of eqs.(2.269) and (2.270) are omitted in
what follows.
Let us now. derive the expressions for determining the coefficients h and s:
= h, + tkb,-^ta, . ^^'^^^^
* ' 'con '
s=5,+%=s.-/Aa, + ^(DiTi+D2x)= (2.272)
=s^ — ikai-]-ib .
con
IW-
The flapping coefficients of the rotor are
(2.273)
(2.274)
For helicopter flight with sideslip, eqs.(2.269) - (2.274) should be cor-
rected. For exanple (Fig. 2. 55) »
fe, =6, + (DiTi + D^x) cos p, j+ (Dix - DjTi) sin p^ ,,
(2.275)
(2.276)
since the inclination of the cone axis and of the aerodynamic force produced ■by-
deflection of the automatic pitch control is determined along axes related with
the helicopter regardless of the direction of the
velocity -vector. The angles a^, a^, bi, bx, just
as the forces H and S, are the angles and forces
along axes fixed with respect to the direction of
the velocity vector.
If we represent h, and s^ in terms of conpo-
nents lying in the plane of the blade tips
[eqs.(2.207) and (2-208)], we obtain
A = A(K) + ^ai = A(K) + /ai + /(DiX-D2Ti); (2.277)
s=S(K) + /&,=S(K)+<&, + /(£>,Tl+^2'')- (2.278)
Equations (2.269) and (2.270) show that_, for /137
a rotor with a flapping conpensator, cpx ^-^-d cpi de-
pend not only on the angles of deflection of the
automatic pitch control but also on the flapping
coefficients of the equivalent rotor. This sub-
stantially conplicates the calculation of a rotor
with a flapping conpensator since, in determining
the initial data for calculating the equivalent
rotor \g, 9, |j., it is necessary to know the coeffi-
Fig.2.55 Deconposition
of ai and b, into
•^con con
Velocity Axes during
Helicopter FUght with
Sideslip .
cients a^ , b^
However, when any five quanti-
ties characterizing the operation of a rotor with variable pitch are prescribed,
it is always possible to select a calculation sequence (sometimes pre-assigning
several values of X^ oi* 9 and constructing auxiliary graphs) which will contain
all coefficients of forces, moments, and flapping of the rotor.
Let us give a typical exanple. At given t (rotor thrust approximately
equal to helicopter weight), iJ,, %, k, and T], the aerodynamic calculation se-
quence for the rotor can be as follows: Assigning various values of \^ , the
expression obtained from eqs.(2.157)"" and (2.242)
* We recall that cpo entered the formulas in Subsections 3-24. For sinplicity,
the subscript "0" of cp was omitted.
145
<Pe=^0 = - ^ ,
1 +-^(1+1^2)
4
will yield cpoj after which eq.(2.50) will give t^ = t.
After deterndning, either by trial and error or graphically, the values of
Xe and cpo at which t is equal to the prescribed value, eq.(2.40) will furnish
ao, ai^ and bi^ ._ Then, ^qs.(2.^9), (2.250), (2.269), and (2.270) will be used
for determining ai, bi, cpi, and cpi . We now have all data necessary for calcu-
lating the characteristics of the equivalent rotor and their conversion in the
axis of a rotor with variable pitch. To detemdne h, s, and m^ we can also use
^formulas derived in Subsection 24, obtaining - in the above-described sequence -
q'i(k) > 9i(k) > ^^d M)!.) ^^°^ eqs.(2.222).
The aerodynamic calculation and the calculation of helicopter balancing are
performed in the same sequence.
As shown in Subsection 23, the aerodynamic characteristics of a rotor in a
velocity coordinate system - t^ = f(ty, m^) at constant values of |j, and Mq - do
not depend on cpi and ^i ; therefore, the conputation can be performed from the
characteristics of a rotor with constant pitch: t^ = f(ty^ , m^ ) for the same
values of |j, (p. = V) and Mq . From the aerodynamic calculation, we obtain the co-
efficients tjj and ty ; at any value of cpi and cpi in a given flying regime, the
characteristics of the equivalent rotor" will not change and will correspond to
the found values of |j,, tx , and ty . Thus, as a result of the aerodynamic
calculation, we will obtain all characteristics of the equivalent rotor. /138
After this, we calculate 9o = cpo + kao, % , bi and, from eqs.(2.27l), (2.272)
or (2.277), (2.278), the components h and s which do not depend on a.^aon ^^^
bx . From the condition of helicopter balancing, i.e., from the condition of
equating to zero the longitudinal and transverse moments, we find ax and bx
and follow this by calculating k , J\, ax , bx , h, and s .
Section 3. Momentum Theory of Rotor
In the momentum theory of a rotor, the aerodynamic forces and the power re-
quired by the rotor are found by applying general theorems of mechanics to the
flow around the rotor.
This theory is used in approximate calculations in which both the induced
"- The characteristics of the equivalent rotor for ar^r cpx are the same when as-
suming that the parasite drag of the fuselage (and also the angle of attack of
the wing of a winged helicopter) does not depend on cpx and is determined as some
average value (9x)av[Q' = c^s - (9iav)]« Usually, this assunption is valid. If
it is not, it will be necessary to perform second-approximation calculations
from the value of ^x obtained from the balancing calculation.
146
and profile power of the rotor are determined on the basis of sinplifying as-
sunptions or from precalculated graphs.. In this case, there is no need to de-
termine the angles of attack and elementary aerodynamic forces in each blade
section, a fact responsible for the sinplicity of the formulas.
In the momentum theory, the con^jonents of the aerodynamic forces of the
rotor along the velocity of flight (drag)
and normal to it (lift) are determined,
which makes this theory convenient for
use in helicopter calculations.
1. Theory of an Ideal Helicopter Rotor
When creating lift and drag (or
propulsive force), the rotor thrusts an
air mass downward and forward (or back-
ward) .
Glauert postulated that the rotor
acts on an air mass passing through the
area of a circle placed normal to the
flow incident on the rotor. The diameter
of the circle was to be equal to the
diameter of the rotor (Fig. 2. 56). This
postulate is based on the fact that the
same flow boundaries are selected both
for the propeller and for the wing, with
uniform induced velocity distribution.
For the propeller, this is entirely ob-
vious since the flow boundary is deter-
mined by the area swept by the blades;
for the wing, the possibility of select-
ing such a flow boundary is given by the
vortex theory. Recently developed vortex
theories of a rotor rather- accurately /I39
confirm the correctness of Glauert 's
hypothesis concerning the air mass participating in the generation of the aero-
dynamic forces of a rotor.
In the ideal rotor theoiy (Ref .21), it is postulated that the air flows at
the same velocity over the entire area of the circle: The air stream does not
mix with the surrounding air, so that it is proposed that the air is an inviscid
fluid. Fiorthermore, it is assumed that profile losses of power and vorticity of
the stream are absent.
V^-V-Vx^
Pig. 2. 56
Model of Airflow around
Rotor .
A model of the airflow and its velocity coirponents in three sections - far
ipstream of the rotor (section O-O), along the rotor axis (section 1-1), and far
downstream of the rotor (section 2-2) - are shown in Fig. 2. 56. The induced ve-
locity corresponding to the rotor lift Y is denoted by the vector Vy, while the
induced velocity corresponding to the rotor drag X, is represented by the
vector Vx ( Vy W^f, v, \\^) and the velocity of the undisturbed flow, by V. All
147
vectors are shown for positive direction, with the subscripts corresponding to
the section number.
From the theorem of moment of moment\mi follow the relations
where m is the air mass flowing per second through the section 0-0, 1-1, or 2-2.
The variation in kinetic energy of the per-second air mass is
E=E,-Eo = -^m{Vl+vl~V^). (3.2)
Equating eq.(3.2) to the expression of energy inparted to the air by the
rotor in unit time
(3.3)
we obtain
After transfonnation, this es^jression is reduced to the form
which indicates that, in the examined flow, the following conditions are satis-
fied:
^i/. =-77 '"y
(3.4)
2
and
T'^^'- (3.5)
■^•^. =^r '"•'.•
These relations show that the induced velocities in the rotor plane are
one half those far downstream of the rotor.
The power supplied to an ideal rotor is e^qjended only for creating kinetic
energy of the flow and thus is equal to it. Making use of eq.(3.3), we have
75N=V-Vy,-XiV~VjcJ. (3.6)
The weight rate of flow of air m per second is equal to the product of the
mass density and the volume rate of flow of air per second:
m=QFV', (3.7)
where V' is the resultant of the velocities V, Vy and v^ :
148
V' = YV]+ vl=y(V-v,,f-\-vl.
(3.8)
To account for the so-called tip losses of the rotor, the following /1Z^0
method is used: In calculating the forces and induced velocities, the air mass
flowing through an effective area F^tt = B^F smaller than the area swept by the
blades is introduced. Usually, for forward flight regimes we take B^ =
= 0.94 - 0.96 [the effective radius of the rotor R^tt = BR = (0.97 - 0.98)R].
The power is calculated on the basis of the mass of air flowing through the
actual area. Therefore, by means of eqs.(3.l) and (3.4) to (3.8), with consid-
eration of tip losses, we obtain
75N=:^iyv,-X(V-v„)].
(3.9)
(3.10)
(3.11)
Let us now change over to dimensionless quantities - coefficients of forces,
moments (power), and velocity:
m
\—^[(^y'^y-Cx{V—vX
(3.12)
(3.13)
(3.1!j-)
In these expressions we omitted the subscripts since, from now on, we will
be concerned only with the velocities in
section 1-1 and, for sinplicity, will not
give them a subscript.
Equations (3.12) to (3.14) describe the
general case where any aerodynamic system
creates lift and drag (or propulsion) and
consumes or yields power. Therefore, these
expressions also are valid for a propeller
and for a wing. For a propeller, Cy = Vy =
= must be substituted into eqs.(3.l2) to
(3.14) and for a wing which does not inject
engine power into the flow, m^ = 0. (For a
wing, one usually takes "v, < V, Vy < V. )
Fig. 2. 57 Deconposition of the
Resultant Aerodynamic Force of
a Rotor into Velocity and Body-
Fixed Axes.
In the rotor theory, it is conventional
to use quantities in rotor-fixed coordinates
Cj, C^ , [J*, A., etc.
Introduction of the angle of attack of the rotor yields the following rela-
tions between the coefficients of forces and velocities, in different coordinate
systems (Figs. 2. 57 and 2.58):
li..9
Cj-^Cy COS a-\-Cx sin a;
V( = "Wy cos a -f "Ux sin a;
Cu=Cx cos a — Cy sin a;
■Wft = 1)^ cos a — -Wj, sin a ;
F'' = (Fcosa-^J + (F sina -^^j'^^a+X^;
\i-^V zosa~v^\
X=: Vsina — z;^.
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
Substituting eq.(3.l2) and (3.13) into eqs.(3.15) and (3.17) we obtain
formulas for the coefficients Cj and C^ :
m
t=— -^(^7->- + C//P')-
iS2
(3.22)
(3.23)
{3.2h)
(3.25)
Let us now study the velocity polygon of a rotor and derive a number of
additional relations facilitating
the calculation of rotor charac-
teristics.
The velocity and force poly-
gons are shown in Figs. 2. 58 and
2.59. The latter diagram, as a
supplement to Fig. 2. 58, shows the
vector of the restiltant induced
velocity
--'Vy-\r'"x='"f\- '"h
Fig. 2. 58 Velocity Polygon of Rotor.
as well as the angles C a-nd 5, as
follows :
C, - angle between the ve-
locity of the undis-
turbed stream (flying
speed) and resioltant
velocity in the rotor region, £ > at Cy > 0;
6 - angle between the normal to the velocity of the undisturbed stream
and resultant aerodynamic force of rotor.
150
Since Cr || u, the angle between the vectors Vy and u vdll also be equal to
6.
The angles C and 5 and the flying speed V conpletely determine the velocity
polygon. To determine the velocity polygon in terms of vectors in a fixed co-
ordinate system, one more quantity
must be known, such as - for ex- /1^2
anple - the angle of attack of the
rotor a .
Let us write out the main re-
lations between forces and veloci-
ties in the velocity polygon:
£iL=^=Un8;
Cy
Fig. 2. 59 Velocity Polygon of Rotor.
sin C cos S cos (C — -8)
Cy = C/j COS 8,
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
'Uy = acos8, 1
•u_j = asin8. J
Using these relations, eqs.(3.l2) - (3.14) can be written in another form
m
Cy= AB^v.V ^4BW' ^'"^^"^'^ ;
^ ^ cos2(C — 8)
Cj^ = Cy 't«n8;
x(^-g)='^^^-(^-^) = ^ ^^^-(^-
8).
(3.31)
(3.32)
(3.33)
Equations (3.31) to (3.33) are of interest in that the two independent _
variables C and 6 correlate the coefficients Cy, Cx, and mt at ar^ value of V,
which is a consequence of the similarity of the velocity polygons in regimes in
C C
which the ratios -=^ and -~- are equal.
yS V^
Having assigned a series of values to the angles C, and 6, we can find
/Mik
the ratios
m
B^
2 * B^V^ * 7^
i- and construct a graph for their correlation.
151
U1
JO
#
?J^##-^^^^^^^-^
^:'„«^L£
-0.10 —0.05
0.03 „ ,
-O.Oit
-'■''-00
0.13 ";f
0.1S
Fig. 2. 60 Interrelation of Coefficients of lift. Propulsive Force,
Cy
and Torque of an Ideal Rotor
SttS
Wl'
< 1.0
E
At Cy = (C = O), eqs.(3.3l) to (3.33) are not applicable; therefore, in
constructing the graph, we used eqs.(3'13) and (3.14) transformed into
Cx _ ,
i73
B2V2
(3.34)
(3-35)
Such graphs (Figs.2.60 and 2-6l) are convenient for solving problems of
aerodynamic design.
~= 0.5 m ^ :S - 4 -5 -6 J
Fig. 2. 61 Interrelation of Coefficients of lift. Propulsive
Cy
Force, and Torque of an Ideal Rotor
B^'^P
= 1.0
- 12.0).
To determine the quantities entering the velocity polygon of a rotor, we
must know - in addition to V - angles C and 6 . The angle £ is determined rather
accurately by means of graphs (see Figs. 2. 60 and 2.6l), while the angle 6 is de-
n
termined from Cy and Cxt 6 = tan" " .
153
At small C and 6 (large V, small Cy ) eqs.(3.3l) to (3.33) are sinplified. /1^5
Actually, we have
Cy=452K=C;
(3.36)
Substituting the first two equalities into the third, we obtain
m
t— — (CyVt:. - CyVh)^^ ~ J- CvF.
^82"-^ y ' 4B4K 52 '"^'^*
(3.37)
At small angles of attack
^ C;,K.
(3.38)
The e^^ression for m^ is generally used in this form in aerodynamic calcu-
lations of a helicopter in fljring regimes at V s 0.15.
To calculate fljd-ng regimes with large £ (at small V), which are not
covered by the graphs shown in Figs. 2.60 and 2.61, we must use eqs.(3.l2) to
(3.14). Substituting into eq.(3.13)
- - Cy
^ Cx
(3.39)
and transforming, we obtain
/Cx\2_2^
\4B2 ,
)=f [(^-^J+1/ (v^-^.r+
4B2 J
(3.40)
This expression permits constructing graphs for the aerodynamic character-
istics of a rotor Cx = f(mt')> ^or any selected values of V and Cy: Assigning
Vx, we find Cx, then Vy, and finally Ef, from eq.(3.14) (the sign of Cx coincides
with the sign of v^ ) .
To calculate the characteristics of an ideal rotor, we can also use the
following expressions:
C^ = 4B^uV' = 4B^u ]/(l/-usin8)^+(«cos8)^ =
^AB'^u]/ 1/2 -a (21/ sin 8- u).
Let us then transform eq.(3.4l) into
(3.41)
154
1 =
4B2
Cr
4B2
■/■
Cr
4B2
2sin8
/
Cr
4B2
(3.42)
from where it is obvious that we can construct the graph of the dependence of
u =
V.
on V
i-
and 6 . Such a graph, borrowed from another publi-
4B^ ' 4B^
cation (Ref .2), is shown in Pig. 2. 62. In this diagram, the broken line is the
approximate curve which can be used as basis for calculating the vortex-ring
state at 6 = +60° and 6 = +90°-; for this curve, the ideal rotor theory does not
hold.
A determination of the induced velocity of the rotor is of independent
interest .
If the angles C a^nd 6 are known, the ratio of the induced velocity to /146
the flying speed can be determined by expres-
sions derived from eqs.(3-26) and (3.27):
u
2.0
19
1.8
n
1.6
1.5
t.'i
I.J
1.2
1.1
1.0
0.9
0.8
0.7
OS
0.5
0.1
0.3
0.2
0.1
/
1
-_i
J
/
/y
1/
\
1
1
-
—
-
y
S-^QO"
^60°
^30°
1^15°
^^
^n<>
^
1
V
\
-^
\
^
a
-^
v3^
%
\\Z^-60°
^1—
>}=-go-
^
1^
—
—
^
'Ni
^
1^
_^
V
sin C cos B 1
cos(C-B) 1 '
+ tan8
tonC
(3.43)
i;^ = 'Uj,ta>i8.
{3.kk)
If the angle of attack of the rotor is
known, the velocities Vt and Vjj can be ob-
.tained from eqs.(3.l6) and (3.18).
In the operating condition Ox = v^ = 0,
the velocity Vy is determined from the ex-
pression:
"'-"/i/t+Ij^-T-t- (3-
45)
At small angles C a-nd 5, the relation
"v-y. < "Sf ±s satisfied, so that
Cy
1.0
2.0 3.0 1.0 5.0 V
or
Fig. 2. 62 Induced Velocity u
as a Function of V and 5 .
1),=
Ct .
4B2 ]/ 1/2^.^2
Cx
4B2 Yv^ +~vl
(3.46)
(3.460
(3.47)
155
At V ^ 0.15 and at small angles of attack (more acciirately, if V ^ 2.0 and
at small 8 where - in conformity -with Fig. 2. 62 - u « -s=-) the sinplified expres-
sions widely used in the calculations will rather accurately yield the induced
velocities
- - Cy - Ct
■u = z;„ = — s.- — 1), - - —
4B2V 4B2(. (3^^g)
(3.49)
In these cases, the induced velocity is denoted by the letter v without
subscript, for sinpHcity.
2. Derivation gX t he Eajression for the Torcpae Coefficient
of a Real Rotor
Equations (3.1^) and (3.37) were derived above for determining the torque
coefficient of an ideal rotor, when considering the rotor as an active disk in-
fluencing its own circumflow. These e^qjressions are interesting in that m^ is
represented as an explicit function of the coefficients_of Hft and propulsive
force Cy and Cx . In the same foiro, the expression for m,. can be derived also /147
for a real rotor. This derivation was originated by L.S.Vil'dgrube. The ob-
tained equation is valid for nonuniform induced velocity distribution over the
rotor disk and takes into account the forces of the profile drag of the blades.
As is known from the classical theory (Sects. 2 and 4), the conponents of
the dimensionless velocity of flow past the blade sections (at l-^^^ =0,
cos p « 1) are equal to
t7,=-^=r + 7cosasin<|-; (3.50)
(7v = -^=Fsin a-ti — Fcosasln8cost>-r -^. (3.51)
^ <»R ^ d^
The con53onents of the aerodynamic forces located in a plane perpendicular
to the blade axis are expressed by the equations
dt ={Cy cos ^ -\-c^pSinO)U''bdr', (3-52)
rf9 = (c^pCOs«)-CySin«))t7^6afr. (3*53)
Substituting, into eqs.(3.52) and (3.53), the e3?)ressions
sin 0) = -^andcos© =-^^ ,
X.J- • U u
we obtaxn
dt=={r.p^^c^Jjy)Ubd'r; (3.54)
156
dq = {c,^U^-c^U,)Ubdr. (3.55)
Solving eq.(3«54) relative to CyUbdr
^ r;Tw " <" C:cfiyVldr
and substituting this into eq.(3.55)» we obtain
dq=c,i^,-\-^Ubdr-^Jt = c,^^%d7-dt -^. (3.56)
The elementary torque of the rotor is
dmt = dqr. (3.5?)
From this, after substituting dq from eq.(3»56) and r from eq.(3.50) we obtain
dm.^ = c^p^Fdr—dtUy — dqVcosas\n<\K (3.58)
For further transformations of eq.(3»58) we use eq.(3.5l) and the e^qares-
sions for the elementary longitudinal and propulsive forces:
dh=:dqsin<if — dtsin'^cos'!f; (3-59)
dt^=:dhcosa-\-dtsina. (3.6o)
As a result we obtain /1Z(,8
Li.VV Oil
4Z ''P^
dmt = c^pU^bdr-\- dtv-\-dtr -^ — dtV s\na+dts\n pcos4>\/cosa-
dqsXn'iiV cosa = c^pU^bdr-\-dtv-\-dtr-^— rf/Vsina— (3-61)
d^
dl
d^
We then integrate the elemental^ torques with respect to the rotor area
~dhVcosa = c^/J^bdr-\-dtv-\-d{r-^-dt^V.
2t 1 2x I 2it 1 2it 1 2ic I
ntf. = frft r rfm^ = r f c^^'^dFd^ -{-U'Sf^dtv -\-\^{ dfr ^d,], _ I? frf^ f ,
00 00 00 00 oo
The equation of blade flapping has the form
157
so that I dtr is proportional to the sum — -§— + p - const. Therefore,
J di|f
2it 1
+jP«fp-constJrfp=i(^y| +± H _constp| =0.
'o 'o
dB
since p and ,P at ij; = and \|r = 2n have an identical value [V.E.Baskin gives
such a derivation of eq.(3.63)]«
Thus, the expression for % can be represented as
27t 1 _ 27t 1
m^^llc^/J^bdrdif -^] d'ifl dti-t^V. (3-64)
For sinplic.ity, we derived the expression for m^ on the basis of eqs.(3«50)
and (3.51) for U^ and Uy . More accurate expressions, taking account of the com-
ponent of induced velocity v^ ,
t7^=7 + (Fcosa-'Uft)sin^; (3.65)
Z7, = Fslna-z)<-(i7cosa — t)Jsinpcos<l<-r ^, (3.66)
will yield, after analogous conputations, the following expression for m^:
/w^ = J J c.^bO^ d7d^ + S^d^l {dt^, - dt^,) -t,V. (3.67)
00 00
The first integral in eq.(3.67) contains the forces of profile drag and the /1^9
second the forces of induced drag. We designate them, respectively, as
2it 1
2it 1
For an ideal rotor for which c^p = and for which the induced velocities
are loniformly distributed, we obtain from eq.(3.67) with an approximate consid-
eration of tip losses,
158
or, multiplying both sides of the equality by the loading factor,
mt=^[CyVy~C^(V-v,)]. (3.68)
Equation (3.68) coincides with eq.(3«14) obtained in the ideal rotor
theory.
We note that in the expression for m^ used for calculations in the classical
theory, the term taking account of the profile drag of the blades does not
coincide with that obtained above and, in conformity with eq.(3.55) and (3.57)»
is equal to
2it 1
m-p,
pr = f f CjcpU U / b dr d^.
The discrepancy of these expressions is due to the fact that the profile
drag forces enter not only into the expression for _ ■- but also into the ex-
pression for — ^ and thus into t^ and ty as well; if c^p is taken into account
dr
at some fixed values of the angle of attack a and the pitch cp of the rotor (i.e.,
true angle of attack of the section as is done in calculations by the classical
theory), then both m,. (by an amount mp^ ) and t^ and ty also will change. When
calculating mt by eq.(3.67), the term mp^ determines the increment in m^ , pro-
vided identical values of t^ and ty are maintained, which obviously occurs at
different o- and cp for rotors with different c^p .
Since it is of greater interest to conpare rotors with different profile
drag of the blades at identical t^ and ty, the profile losses of the rotor are
estimated with respect to the quantity mp^ calculated at angles of attack and
pitches of the rotors corresponding to the same value of the coefficients t^
and ty .
For the reasons presented a.bove it is obvious that, for changing from an
ideal rotor with certain ty and t^ to a real rotor with the same ty and t^ , the
profile losses must be determined from the expression for mp^ ♦
It also follows from eq.(3.67) that the influence of a nonuniform induced
velocity distribution over the rotor disk at given ty and t^ is directly deter-
mined by the quantity m^^^ . Fiirthermore, the form of the induced velocity dis-
tribution influences the angles of attack of the blade sections and thus also
the quantity mp, .
The flapping angle of the blades does not directly enter into the co- /150
efficient m^ owing to the fact that the integral (3.63) is equal to zero. How-
ever, flapping does influence the distribution of c^p , dt, and v over the rotor
disk and hence the quantity mp^ and mi„4 .
159
3. Rotor Profile Losses
As shown above, at given values of the coefficients ty and t^ the rotor
profile losses are deteraiined 1^ the expression
2x 1
'"^"^n ^'"^^^ ^'' ^^- (3.69)
With consideration of the radial velocity conponent of flow past the blade
(see Fig. 2.91), the profile losses are determined by the e^^jression
2it I
m^r = f J c^p (U^ + K2 cos2 a cos^ <^fi^ bdrd^i. ( 3 . 70 )
Equation (3 '70) should be used at small values of Mq and ty , i.e., in
cases when the coefficient c^p is determined mainly by friction forces. At
large Mq and ty , when regions of flow separation and an increase in wave drag
appear on the rotor, the profile drag of the sections is determined by the ve-
locity conponent of the stream U normal to the blade axis, and eq.(3.69) must
be used for calculating mp, .
To calculate mp^ it is necessary to know the distribution of the profile
drag coefficient c^p and velocity of flow U about the blade sections over the
rotor disk. Consequ.ently, a calculation of mp^ is a laborious task and, in
practice, can be performed only on high-speed conputers.
Figures 2.63 - 2.70 give graphs of mp, as a function of the coefficients
■ty> tx, V, Mq = —^ — for a rotor with rectangular blades and a loading factor
of a = 0.091. The rotor blades (variant II in Table 2.10) have a linear geo-
metric twist Acp = 7°» The blade profile is as follows: at the shank, up to r =
= 0.85 - NACA 230 with a relative thickness 'c = 12^, at the end of the velocity
profile with a relative thickness "c = 9^« The coefficient Cxp obtained on ex-
posing the model to an airstream increased by Ac^p = 0.002 and pertains to a
rotor having a high profile drag owing to poor manufacture. The aerodynamic
characteristics of the profiles are given in Section 4, 3«
The calculation was carried out by means of eq.(3'69)- Here, it was as-
sijmed that the induced velocity is constant over the rotor disk and was deter-
mined by eq.(3«46).
The method of calculation and the remaining assunptions are described in
Section 4 '2 and 4«4»
We see from the graphs that at numbers Mq =0.6-0.7 the quantity mp^
greatly depends on ty and t^ •
At Mq ^ 0.5, the quantity mpp depends little on tjj and increases somewhat
upon an increase of ty .
160
nM
Fig.2.63_ Coefficient of Profile Power of Rotor
(V = 0.15; Mo = 0.6; a = O.O9I).
t^
1
\
\
' '-
^ —
I
■~
\.
\
\,
\
^
_
f
,
^
\
\
>"
|. -
\
\
\,
\
1
S
0.011Z
1
om
\
\
0.0
01
\
0.005
O.OOB
m.pr
ty-O
OS^].
\
90.-5'
A fl*
. -J*
A -10'
X -15'
\
\
V
s?-
,
\
\
\
h
3
—
/7/77
^
^
>
,■=;
\
\ w
v?
\
\
Ir
\?
=1
^
\
^_
\ 1
Fig. 2. 64 _Coefficient of Profile Power of Rotor
(V = 0.2; Mo = 0.6; a = O.O9I).
161
t>
A
—-
—
—
—
1
'
V '
1
'
--
1
1
QOOJi
-
a
,
'
lycw
t,-c.ot
1
\ t^O.Oi
'\
\
I '^
oov\
O.0OS
"V/-
\
\
—
1 •
1
, \
*,
1 Ol-O
• -/♦
« -«•
« -«•
\
\
V-
-
- \
\
\
\
\
/
\ '
\
\
-
tAo
'n
\
—
1/
-ty=^.l
1
rt^V
\
1
\
1.
Zi^
(T^/
-OjOl
-0.02
A
^
A0£-
■0
—
••o
-1
• -5'
A -to'
« -«•
tr -20'
- t,-o.
?»-
\
\
T
\
aooull
<?W5
\
0,000 m
\
V
\
\ \
Vx
\
\
V
^^1
'\
\
\
\
J
^\
N
V,
\\
\
(f
\
\
h
If
\
\^
\
Fig. 2. 65 Coefficient of Profile Power Fig. 2. 66 Coefficient of Profile Power
of Rotor (V = 0.3; Mq = 0.6; a = 0.091). of Rotor (V = 0.4; Mq = 0.6; o = 0.091).
tx
~^
r
~'
r
^
■> _
\
\
s
\
\
L.
\
■
\
s
\
1
\
\
\
y
\,
0.002
0.003
\
0.001^
0.005
\
K
0.0
06
0.007
mpr
'
1
\
1 1
\
\
^,
ort-5*
ty=O.0B'
4
\
c^
\
\>.
A
• -r
* -w
I
•*1
r*-
—
\
t
\
\
NT
\
«h
V
^
\
X -15'
\
^\-^
r
\
\
P -20'
\ \
\
\
J
N
L
[_
J
Fig. 2. 67 _Coefficient of Profile Power of Rotor
(V = 0.15; Mo = 0.7; cr = 0.091).
162
/l^
ix
0JJ2
-0JI2
I
^
\
\l 1 1
4<
■
\
-■
\
%
^
\^
\
■\ 1 i
^1
nP^\
\ejm
6.025
o.oot
mpr
1
\
\
ort-^
-°
\~^r
'0.06
\
\\ 1
• -^
f
\
\
* -
0"
yi I rt
\
1*
X
%
- -
'r
i'
l>^^'*
^t.!!
1
: _t^r
V.
^\•
X7
\*
s
?\ 1
i\
\
Fig. 2. 68 _Coefficient of Profile Power of Rotor
(V = 0.2; Mo = 0.7; CT = 0.091).
U.Ui
t
\
0.0014 '
t
\
. \
\
\ ^
v
\
\
^
, \
V
1 \
\^
\ \
\ o.dor^
0,006 \
mpr
\ x*^
\
r T^
• -5
\ \^^
'OJJi
X X
X -/J
-2t
r
1 w ^
A -to
o
i;3'
\
t^O.OB *-
1 1
r i^
^
k
\
^..
Fig. 2. 69 _Coefficient of Profile Power of Rotor
(V = 0.3; Mo = 0.7; o- = O.09I).
163
0.01
-0.01
-0.02
A
---
\~
-
—
--
\
\
\
—
1
1
r
\
0.007\ t\}
0.008
0.009
^^.oto
mpr
'
\
fl' 1
*
V^
\
■i=.t
k
' \
\
\\
^
—
3
\-i
• -5° -
» -w —
» -!5'
-20° ~
o -25'-
1 1 1
?
'
s
\
A
\>
ifd.06\
3
s.
\f
\
\,
\
i
1
\
J
The materials for determining
Mpj. for blades with other geometric
characteristics are presented below
in Subsections 4-6.
/154
4« Certain Considerations in Selec ting
Blade Shape and Profile
The power expended to overcome
profile losses of the rotor consti-
tutes a large portion of the total re-
quired power of the helicopter. As
shown in Fig .3. 52, about 50% of the
required power is e:!q3ended for over-
coming the profile drag of blades in
horizontal fUght.
Fig. 2. 70 Coefficient of Profile
Power of Rotor (V = 0.4; Mq =
= 0.7; cj = 0.091).
Since, the induced losses consti-
tute a smaller portion of the losses
than the profile losses, we can con-
sider that the conclusions as to the
effect of the geometric characteristics
of the blade on profile losses pertain also to the total power of the rotor,
especially at high flying speeds when the induced velocities are small and the
induced losses do not exceed 12 - 15% of the total power.
Figures 2.71 - 2.74 contain conparative graphs of the profile drag coeffi-
cient mpr for rotors with five blade variants:
Variant I - trapezoidal twisted blade with high-speed profile at tip;
Variant II - rectangular twisted blade with high-speed profile at tip
(rotor described above);
Variant III - rectangular twisted blade with NACA 23012 profile;
Variant IV - rectangular twisted blade with symmetric NACA 0012 profile;
Variant V - trapezoidal flat blade with high-speed profile at tip .
A detailed description of all blade variants is given in Table 2.10.
A conparison of the blades is carried out for average and large lift coef-
ficients, at two values of V: 0.2 and 0.4.
A conparison shows that at low Mach number M^ :S 0.5 the trapezoidal /I55
twisted blade, at all values of the propulsive force, has approximately &% less
profile power losses than the rectangular twisted blade. Since, in horizontal
flight, about one half of the required power is expended to overcome the profile
drag of the blades, a decrease in mp^ by 8% will lead to a decrease in the re-
quired power coefficient m^^ ^ by 4%«
Therefore, for light helicopters for which Mq is small and \ax " 0.3, the
optimxmi planform of the blade is trapezoidal.
The plane blade in an autorotation regime does not differ from a twisted
164
-0.01
-0.02
-0.03
0.005 mpr
Fig. 2. 71 Coefficient of Profile
Power of Rotors with Blades of
Different Shape (V = 0.2; Mq =
= 0.6; a = 0.091).
blade, but becomes appreciably worse
than the twisted blade in helicopter
regimes, especially at large V. It can
be used for a helicopter only at small
and average values of ty and Mq < 0.6.
•When varying the blade profile at
small peripheral velocities, mp, will
vary within 5 - 1.2%. The symmetric
profile is somewhat better than the
asymmetric; at V = 0.2, the blade with
a thin high-speed profile on the tip
has smaller losses.
We should mention that the influ-
ence of the quality of manufacture of
the profile on mp^ can prove to be
greater than the effect of the type of
profile: Amp^ for different profiles
is about 0.0002 (the maximum difference
at large ty is not more than 0.0004),
whereas owing to difference in the type
of construction and quality of manu-
facture of the blade the profile drag
coefficient of blade sections may dif-
fer by an appreciable amount going /157
as high as 0.003 - 0.004 (see Sect .4,
3), which gives a difference in pro-
file losses - in conformity with
eq.(3.71) - of
Awz,
0.003
/"•-
(1+3-0.32)=^ 0.001.
At high Mach number Mq (Mq = 0.7; tuR = 230 - 238 m/sec), the use of a high-
speed profile at the blade tip markedly reduces profile losses .
The decrease in mp^ amounts to 0.0015 at V = 0.2 and 0.004 at V = 0.4.
This reduces mp^ by 40 - k5% and the total required power, by 20 and 25^ re-
spectively.
In flying regimes (t^ < O) of helicopters, mpp is greatly affected by the
geometric twist of the blade. A straight blade is not used in helicopter
regimes, and in autorotation regimes its profile drag does not differ from that
of a twisted blade.
The trapezoidal blade is better than the rectangular one in autorotation
regimes and at low propulsive force. In helicopter regimes where the angles of
attack of the tip sections of the trapezoidal blade are larger, the difference
in mpi. decreases while at large t^ the rectangular blade becomes better.
At moderate Mach n-umber Mq (Mq = 0.6; cuR = 197 - 204 m/sec) the peculiari-
165
Zi^
t.
1
\
M
\
1
\
1
J
\
\
I
'i
2?\^
-
i
y
1^1
\_
M
w
aoi
L t
N
I
1
\
\
-
\
\
_^
k
l\
\
\
yi\
\
\
\
\
It
-
*»
\
'
\I
\
4
(WOz
0.00J ,
(2eo« \
o.ooe
\
\
1
0.007
-
mpr
— r
V
\
-
-
\
\
y
1
\
\
\
i
\
\
\
-0S1
\
V I
\.
\
\'
-
n
,
\
\ \
\
\
K
v
s
1 ^ >
\
\
\
\,
%\
\
-
1
1-.
\
^
\^
-
—
nl
H
>
-om
'• -5
' A -10
X -15
t -
iy=0J6
.ty=P.2
\
V
\
\
•
T
\
s^
\
\
-
•
jj
\
s
.\
\
\
Yv
\
1
s
N
\
-
\
\
-
^
\
V
\
-BOX
S
^
J
^
K.
>3v
Fig. 2. 72 Coefficient of Profile Power of Rotors with Blades
of Different Shape (? = 0.2;Mo=0.7;<J= 0.091).
-OSII
-<uoz
Fig. 2. 73 Coefficient of Profile Power of Rotors with Blades
of Different Shape (V = O.4; Mo = 0.6; o = O.O9I).
166
ties of the curves of mp, , which were noted at Mq =0.7 (to a larger extent, at
V = 0.1+) t begin to appear: The twisted blade with a high-speed profile at the
tip hecomes better, and the straight blade in helicopter regimes becomes ap-
preciably worse than the twisted blade.
A conparison of rectangular and trapezoidal blades for Mq = 0.6 - 0.7 shows
that at V = 0.4 and also at V = 0.2_for large ty the former has the advantage,
whereas for medium and small ty at V = 0.2 the trapezoidal blade becomes of ad-
vantage. In general, the rectang-ular blade is preferable for heavy and mediimi
helicopters, whereas it is preferable to use trapezoidal blades for rotocraft
for which the coefficients ty and t^ of the rotor are small at large V, owing to
the installation of a wing and a tractor propeller.
-OM^
Fig. 2. 74 Coefficient of Profile Power of Rotors with Blades
of Different Shape (V = 0.4; Mq = 0.7; o = 0.091).
At V = 0.4, the profile losses are quite large even with a high-speed /158
profile at the blade tip: mjjj. is twice that at Mq = 0.4 - 0.5« To estimate the
possibility of decreasing the quantity mp^ , I'igs.2.75 and 2.76 give graphs of
for blades of the variants I and II and also for a rectangular blade with an
nipr
increased geometric twist (variant VI), for an expansible blade (T] = 0.5; vari-
ant VII), and for a rectangular blade with an increase to r = 0.75 of the part
with a high-speed profile (variant VIII). We see from Figs. 2. 75 and 2.76 that,
in horizontal flight and especially in autorotation regime, the trapezoidal
blade remains preferable. At large values of ty , the optimum blade is the blade
with increased twist, which reduces mpj. at V = 0.2 by 20^ '■"^•th.f ^^ -'-'-'^) ^^ ^*
V = 0.4 by 10^ (mt by 5%). Consequently, the use of a blade with greater
twist raises the dynamic as well as the static ceiling of the helicopter, in-
creases the static thrust (see RLg. 2-171), negligibly increases the maxim-um
167
H
On
-0.0t
-m
Fig. 2. 75 Coefficient of Profile Power of
Rotors with Different Blade Shapes
(? = 0.2; Mo = 0.7; CT = 0.091).
Fig. 2. 76 Coefficient of Profile Power of
■ Rotors with Different Blade Shapes
(V = 0.4; Mo = 0.7; a = 0.091).
speed, and appreciably increases the rate of descent in autorotation. An en-
largement of the blade portion with the high-speed profile slightly reduces the
value of nipr .
The obstacle in using blades with greater geometric twist lies in the in-
crease of dynamic stresses in the blade spar, whereas for blades with an en-
larged high-speed profile, the increase of hinge moments is the obstacle.
The expansible blade is preferable over the rectangular design only at very
large values of the propulsive force coefficient tx •
A comparison of the graphs of mpr for Mo = 0.6 and Mq = 0.7 shows that, in
a rotor with our high-speed profile at the tip, it is impossible to avoid a pro-
nounced increase in profile losses at Mo = 0.7, for all blade variants.
The method of utilizing graphs of mpr for rotors with these types of blades
but with a different loading factor is described in Section 6.
5. Approximate Determination of Rotor Profile Losses
The quantity mpr is most reliably determined from graphs plotted for each
specific rotor. If there are no such calculations, the data of Figs. 2. 63 - 2.74
can be used for an approximate estimate of mpr.
At small Mo, the approximate equation (3.72), derived on the assiimption of
constancy of the coefficient cxp in all blade sec-
tions, can be used for determining mpr.
Let us derive eq.(3.72). For a rectangular
blade, we have b = const = 1.0. Having taken
TABLE
2.5
1
1
2
3
4
p
1.0
0.94
0.91
0.88
U^ = {Ul + UlY'^^{r+Vsmif)\
we obtain
m
U • U
Wdr =
(3.71)
where cxp,^ is the average value of the coefficient Cxp over the rotor disk.
For trapezoidal blades, mpr is smaller than for rectangular blades. This
is taken into accoimt by the coefficient P which is pre-assigned in relation to
the blade taper T) (Fig. 2.77) in Table 2.5.
To account for the influence of the radial velocity component of flow past
the blade, mpr is calculated by eq. (3.70). An approximate estimate (Ref.25, 36)
shows that, to account for this component, the coefficient of V^ in eq. (3.7I) /I6O
should be changed from 3 to 5.
169
Axis of rotor rotation
ih
V//M/d
n-i
"tip
tip
Fig. 2. 77 For Determining Blade
Taper.
Thus, the final formula for detei^
mining mpr reads
^p^=\^.p.^^■^^^')p■ (.3.12)
The average profile drag coeffi-
cient Cxpav over the disk is determined
as a function of the average lift coef-
ficient CyQ over the disk; the latter
is found from eq. (3«74) whose deriva-
tion is given below:
2ic R
1-K 1
= -^ QCy, bR {y>Rf -1 J rf^ J t72 cfr.
Substituting
t/2;^(r + Vsin<l;)2
and integrating, we obtain
(3.73)
Expressing Y in terms of the dimensionless coefficient Cy or ty, we find
Cy 3 3 ^y
"■yo-
<s 3 _ 3 -
1-1--— y2 l-f V2
2 2
(3.74)
Having determined Cy^, the profile polar in the section r = 0.7 will
yield CxPay.
L.S.Vil^dgrube proposed to take into account the planform of the blade by
the coefficient P and to determine Cxp ^y ^-s a function of Cy^ .
6. Effect of Air Compressibility on Rotor Profile Losses
At average and large Mo (for profiles generally used at Mq > 0.55 - 0.6,
i.e., at cuR > I85 - 200 m/sec) it is necessary to supplement mpr, calculated
from eq. (3.72), by the term Amoo which takes into account the increment in
170
profile power produced by the increase in profile drag coefficients of sections
over which the flow has high Mach numbers. Thus, /I6l
m^, = 4-r,„ (l+5l/2)P+Am,
(3.75)
W;
The coefficient of the increment in profile power Amco shoiild be determined
with consideration of the actual distribution of the angles of attack of the
blade sections over the rotor disk,
since an increment in profile drag due
to an increase in Mo generally occurs
in all blade sections. Figures 2.78
and 2.79 show M.M.Tishchenko»s graphs
of the variation in the profile power
coefficient of a blade as a function
of its azimuthal position in the
plane of rotation. We see from
Fig. 2. 78 that, at low flying speed
(V = 0.2) but at large thrust coeffi-
cient, the profile power of the blades
increases at all azimuths as the Mq
increases. At high flying speeds
(see Fig. 2. 79), the increment in pro-
file power occurs mainly at azimuths
of 30 - 150°.
OfitS
0.01
oms
■ 1
MifOl
V-'0.2;t/-0.Z
/
-
<
0.65
/'
^"^
X
^-
K
><
y ■
30
^
^
■^
^
-<J
^
—
—
^
-—
' —
Z::!
wo
ZOO
300
d)'
Fig, 2. 78 Variation in Profile Power
Coefficient of Blade with Respect to
Azimuth.
The graphs of Amoo for the variant II of the rotor are given in Figs. 2. 80
to 2.84. The quantity Amoo is defined as the difference between the profile
power coefficient_mpr at the examined Mo and at Mo = 0. 4 at identical values of
the coefficients V, ty, tx :
A/n,„ (Mo) = m^r (Mo) — /«a^ (Mo = 0.4).
(3.76)
It follows from Figs. 2. 80 - 2.84 that Amoo is a fimction not only of V and
the Mach niomber Mq but also of the coefficients t-^ and tx . The coefficients ty
and tx have an especially strong effect at small V at which, in conformity
with Fig. 2. 78, the increment in mpr occurs at all azimuths. Upon an increase
in V the increment in mpr occurs mainly in the region 'I' = 90° (see Fig. 2. 79)
where the angles of attack of the sections are close to zero regardless of the
value of ty. Consequently, at V = O.4 and V = O.5 the influence of ty and tx
on the quantity Amoo is insignificant.
We see from Figs. 2.80 - 2.84 that, at large Mo, V, and ty, Amco is large.
The quantity Amoo ^eatly increases when Mo > 0.55 - 0.6. At near-separation
values of ty when V = O.I5 and V = 0.2, Amco has a high value already at
Mo > 0.5.
So as to keep the increase in required power of a helicopter, due to the
compressibility effect, from exceeding 15 - 18^, the rotor of the variant II
should be used when Mo =_0.7 at V ^ O.3, and when Mo = O.65 at V ^ O.4. For
example, when Mq = 0.7, V = O.3, ty = O.I5, and tx^^^ = -0.0075» the increment
171
/162
0.035
0.030
0.025
0.020
0.015
0.0)
0.005
V'-O.l; by"0.11
\
\
\
\
\
\
\
1
r
\
1
\
L
.^M 0=^0.7
-
-
1
\
\
_^
^
0.65
\\
^
0.6
Jj
/
'\
j^
■^
^0.55
//
'/
N
^
\
^
y
"^
)\
.--
f
^
U
^
^
i'
j^
--'
100
200
300 (6'
Fig. 2. 79 Azimuthal Variation in Profile Power
Coefficient of Blade.
V'O.IS
)0.W
]troiB
Fig. 2.80 Increment in Profile Power Coefficient of
Rotor, due to Air Conpressibility.
172
Z162
0.001)
1$>0,16
0,1 Mo
Fig. 2.81 Increment in Profile Power Coefficient of
Rotor, due to Air Compressibility.
173
^064
op Mo
Fig. 2.82 Increment in Profile Power Coefficient of
Rotor, due to Air Compressibility.
174
/165
"CO
0.005
0.001
0.003
0.002
0.001
V =0A. G =0,091
J
" r r I T II 1 "
-
h-
--
-
-
-4
1
\ -
.;
i
I
— f ~
h
r_L..
A^
m^-
-j
-
. .
g
^
^
i
'^
—
3\ty-0.tS
\0.1U
0,10
Fig. 2.83 Increment in Profile Power Coefficient of
Rotor, due to Air Compressibility.
175
^'"co
y^Q3; 6='0,091
n
/
009
om
/J
n
0.007
1
1
_
0.00S
0.005
)
f_
OfiO'i
1
—
0.003
1
-
--
/
0.002
/
/
0.001
y
—
—
^
^
_
^66
at/
H5
0.S
VTM,
Fig. 2.84 Increment in Profile Power Coefficient of
Rotor, due to Mr Compressibility.
4a„
-10 ■
-2.0 -
-3.0 .
0.1
0.5
0.6
0.7 Mc
i
-
-~
^^
~T
.
^
ty=0.lS\
1
^
<-/-
u.iv
j
'"N
: I
7=«<l; 6=0091
-
ty^O.W \
1
-
-
-
-
i
Fig. 2.85 Increment in Angle of Attack of Rotor, due to Air
Compressibility at Constant Coefficient of Propulsive Force.
AOlr.
-1.0
-2.0
-3.0
0.Q
0.5
"^
V-O.S;
6=0.091
-
0,6
N
0.7 Mo
\
\
\
Fig. 2.86 Increment in Angle of Attack of Rotor, due to Air
Compressibility at Constant Coefficient of Propulsive Force.
176
in profile power vd-ll be AiUoo = 0.0016 which amounts to IBfo of nit^.f • When
Mo = 0.65, V = 0.4, ty = 0.13, and txj^.f = -0.0133, the increment in profile
power will be Amco = 0.002 which amoiints to 16^ of mtj,. f .
Since, at large 7, the increment in mpr ;jr occurs mainly at azimuths close
to 90 , the increase in Amco at large V is intimately connected with the rela-
tion between the Mach niomber of the blade-
Am
0,001
■0.003
0,002
0,001
Fig.2.S7 Increment in Profile
Power Coefficient, due to Air
Con^sressibility for Rotors with
Blades of Different Shape.
tip section at i^ =
90°,
equal to Mn + Mo =
= Mo(l + V), and the critical Mach number /167
of the section profile. The critical tfech
number Mcr is determined at Q? = since, in
the tip sections at ilf = 90°, we have oTj. « 0.
These data show that a 15 - 18% in-
crease in required power, due to the comr-
pressibility effect, occurs at Mfj + Mo =
= 0.91, i«e., Mfi + Mo is larger by 0.1
than Mcr of the high-speed profile when
or = (see Fig. 2.99). At Mn + Mq = M^r +
+ 0.15, the increase in required power is
about 30^. At Mfi + Mo = Mor, the compres-
sibility effect is Arirtually absent. These
relations between Mq, Mfi , and Mor of the
blade profile can be used when selecting Mq
for a helicopter with high flying speeds.
Since, on an increase in Mq, the angle of attack of the rotor shoTild be
more negative so as to retain identical values of the coefficients ty and tx ,
the graphs for the increment in rotor angle
v=o.«; 6=0.031 of attack are given in Figs. 2. 85 and 2.86:
Am CO
0.008
0.007
0.006
0.005
aoon
0.003
0,002
0.001
tx =-0.010
m
_/
^
ty = O.W
//
ty = 0.t1
/
V
7
/ /
u
h
y.
ri
h
///
n
t
7/
j^
^-
~^t-7^
^^5^
^^i^
_=^=^£ =
0.f
0.5
0.6
07 Mg
Fig. 2.88 Increment in Profile
Power Coefficient, due to Air
Conyressibi li.ty for Rotors with
Blades of Different Shape.
Aa„ (Mo) = a(Mo)-a(Mo=0.4).
(3.77)
No graphs were constructed for V «
= 0.15 - 0,3 since, at all ty and tx, the
quantity Affoo does not exceed 1 .
Figures 2.87 and 2.88 show comparative
graphs of Amco for rotors with blades of the
variants II, III, and V. Calculations
showed that the quantity Amoo is greatly
affected by the type of profile (this is
seen from a con^iarison of the variants II
and hi) and by the geometric twist of the
blade (variant V, straight blade). The
planform of the blade plays a role only at
large ty in which case, for trapezoidal
blades where flow separation begins earlier,
Amoo is greater than for rectangular
blades. The planform of the blade plays a
minor role at large Mo in view of the fact
177
II
that, as will be shown in Section 4«7, a variation in planform will cause a
change in the angles of attack of the sections mainly at azimuths ^ = 250 - 340°
where profile losses are small in the pre-separation regime.
A comparison of blades with a high-speed profile at the tip (variant II)
and without it (variant III) used in the calculation, will show that a high-
speed profile must be established at the blade tip when Mo > 0.6 - 0.625*
The graphs of Afflco and Affoo should be calculated for each specific rotor.
However, if no such calciilations are available, the data in Figs. 2.80 - 2.88
can be used for an approximate estimate of Amoo.
The graphs in Figs. 2.80 - 2.88 are laid out also for taking into account /168
the influence of the Mach ntimber Mq on the aerodynamic characteristics of a
rotor, for cases in which the rotor characteristics experimentally determined
at low Mo are to be used also at high Mq. Furthermore, the graphs are useful
for aerodynamic calculations to avoid interpolation of the Mach mm±ier Mq if the
calctilated Mo values do not coincide with those for which the graphs of the
aerodynamic characteristics were plotted.
To use the graphs shown in Figs. 2. 80 - 2,S3 for rotors with similar blades
but with a different loading factor, it is necessary to recalculate the coeffi~
cient tx for a = 0.091 (Sect. 6).
7. Induced Losses of a Real Rotor
Assuming a constant induced velocity .over the entire rotor disk, the torque
coefficient can be obtained from the graphs in Rigs.2.60 and 2.61. The inter-
dependence of the ratios Cy/B^V^, Cx/B^V , and mt/V°, which was derived in the
theory of an ideal rotor, is valid when these ratios are determined with respect
to the total forces Y and X taken with consideration of the profile drag, since
the forces of the profile drag also create induced velocities so that the ve-
locity polygons and all relations given in Subsection 1 remain in force. We
must add the profile losses to the mt obtained in this manner.
Consequently,
V
where
m,==~^i-V'^m,., (3.78)
The addend in the expression for St, containing the product of the aerody-
namic force and the induced velocity CrU, will be called the induced losses of
the rotor.
In calculating the induced losses, we introduce a correction for taking
into account the nonuniform induced velocity distribution over the rotor disk.
178
As follows from eq. (3»67)» the induced losses of a real rotor are deter-
mined by means of the formula
27: 1 _ 2k I _
"I ind = J ^=^1* j dt^u = J rf^p j {dtyVy -dt^v). (3 . 79 ;
However, to calculate ordinary helicopter regimes, at ty > tx, an approxi-
mate expression is used
m^^^j'^cl<^'^dtv. (3.80:
First, just as in the ideal rotor theory, we determine mi„ii under the as-
sumption of constant induced velocity over the entire rotor disk. With this as-
sumption and with an approximate consideration of tip losses, the expression
for mind takes the sin^jle form
m
For flying regimes at V ^ 0.15, • substitution of eq. (3.4S) for v will yield
mind=^. (3.82:
We will demonstrate that eq. (3.82) holds not only for the assumption of
constant induced velocity over the rotor disk but also for an induced velocity
distribution obeying the law
v{r,'^) = v-\-arcos<\i, (3 •83,
where a is a constant.
According to eq. (3.83), the induced velocity has a minimum value in the
forward portion of the rotor disk (If = tt, r = 1) and increases linearly in the
direction of the velocity flow. In a direction perpendicular to the velocity
flow, the induced velocity remains constant. Thus, the form of the induced ve-
locity diagram is a cylinder cut off by a plane turned toward the plane of rota-
tion of the rotor about an axis perpendicular to the direction of motion (see
Fig. 2. 6); the angle of turn is characterized by the quantity a; v, the average
induced velocity of the disk, is determined from eqs.(3«46) or (3«48)«
The induced velocity diagram described by eq. (3«83) is close in character
to the time-average induced velocity diagram found from experiment (see Fig. 2.3.
Let us substitute eq. (3*83) into eq. (3.8O) and find
2nl__ _2ill 2r. 1_
mjnj= Cdi]) ^ dt{v-{-ar COS <Sf)=v f rftj) frf^ + a ^cos'^id'^ l'^^''-
06 6060
179
The integral in the first addend is equal to the rotor thrust coefficient.
The integral for radius in the second addend is proportional to the sum (see
Subsect.2) — -5— + P - const for a rotor with flapping hinges. With an accuracy
d^B ^
to the first harmonics of flapping, the sum 5- + P = aoj consequently, fdtr
is a quantity independent of the azimuthal position of the blade. Therefore,
2TT X
Jcos ijr d'ifjdtr = 0.
o
Thus, for the induced velocity distribution in accordance with eq. (3.83)
the induced losses are also determined by eq. (3«Sl). Calculations based on the
vortex theory for a rotor with an infinite nxmiber of blades show that, owing to
differences in the induced velocity diagram from eq. (3.83), the induced losses
of a rotor with twisted blades are about 5% greater. Taking B = 0.92, the in-
duced losses of the rotor in flying regimes at 7 ^ O.I5 are determined by the
expression
^,^, = _L:0^ -^^0.285-^. (3.84)
'"° 0.92-4 V V
In flying regimes with small V, the quantity mind is found from eq. (3.85) /170
where u is determined as a function of Cr, V, and 5 from the graph in Fig. 2.62:
— 1-05 — 1.05^3/2 ~ n t;cr>3/C
where, just as in eq.(3»84), I.O5 is a coefficient taking into account the in-
crease in induced losses.
The dependence of u on V can be refined by flight tests. After determining
the reqiu-red rotor power from flight tests for a nianber of horizontal flying
speeds and after calculating the parasite drag of the helicopter, we find Cy^.f ,
C^Xh.f (see Chapt.III, Sect. 1.2; as well as Cr, mt^.f , and then V and u from the
expressions
/
452
-__;%., -'"'"•+i-c^M^
0.56 Cf ■ (3.87)
The graph of u = f (v) obtained from flight tests of the Mi-4 helicopter is
shown in Fig. 2.89.
The tests were performed at different heights between rotor and surface of
the airfield h. In flights close to the groiond, the quantity u was affected by
180
the "air cushion". The values of u in the influence domain of the "air cushion"
are plotted on the graph for different values of h = .
R
The graph of u in Fig. 2.89 was obtained for low horizontal flying speeds,
when S = &^,t « 0. However, the diagram can be used approximately for values
of S within limits from +5° to -30° . . Calculations made from the graph pf u,
shown in Fig. 2.62, revealed that instead of determining the product uCr ^we can
calciilate the product uC? (Cy = Cr cos S), determining u for 6=0 and V =
= 1.96
V
H''7h)^^-"t''-^)
cT.
(3.88)
Thus, in calculating the rotor characteristics at low flying speeds, mina
is determined from the expression
10
<:
^
/
-f
.>
V
-
1
^
0.7S
\J
.\
\
^
\
\^
\
—
N
s
\
4
--
\
S
\^
k
\
\
^
\
^
0.50
N
N
V
—
\
V
^
V
^
^
iSa
1^
0.25
i
-
-
m
ind
= O.SGiiCy-.
(3.89)
t.o
z.o
3.0 V
Fig. 2.89 _Induced Velocity u as a Function
of V and h (Based on Flight Tests of the
Mi-4 Helicopter).
Fig. 3. 8) close to y^ = 0.057, just as for the Yak-24.
Equation (3.89) can be used
both for calculating the rotor
power reqtiired for horizontal
flight at low flying speeds and
for determining the propiilsive
force of the rotor when calcu-
lating the takeoff distance of a
helicopter or the towing force of
a towing helicopter. These
calculations are substantially
simplified because of the fact
that, for determining the veloci-
ty coefficient 7, it suffices to
know Cy and not Cr .
The graph of the average in-
duced velocity u for a rotor
system (with consideration of
mutToal interference) of the
Yak-24 fore-and-aft helicopter is
shown in Fig. 2. 90. Figure 2.90
also contains the curve u for the
Mi-4 helicopter outside the
earth »s influence. This graph
can be used approxima'tely for de-
termining u of all helicopters
of single-rotor configuration /171
and of fore-and-aft helicopters
■with an excess of rotors yv (see
181
Turning to these ciirves, we can find u for fore-and-aft helicopters with
other y and also for helicopters of side-by-side configuration, after determining
u at high flying speeds (V ^ 0.15) from eq.(3.87), based on data of an aerody-
namic calculation. Such a ciu-ve is plotted in Fig. 2. 90 for a helicopter of
side-by-side configuration with a
10
^^
.
-
-
S
^
\
\
\
■-
\
<^
'
\
;>
N
\,
Yak-2^ helicopter
' (fore-and-aft
configuration)
s
\
—
V
<
0J5
\
\
S^
s
N
k
\
\
\
N
k
^
'v
0.S0
\
s.
N
K
M£-4 hel
icopter
\
configuration)
\
k
\
X
1 1 y
Hel icopt
:r o
K
--^
0.Z5
side-by- side
V
(X^^-O.V)
■^
^
—
—
coefficient
Hss = -O.4.
of mutual induction
It_is interesting to note that
at low V for a fore-and-aft heli-
copter the induced velocity coeffi-
cient, owing to the mutual inter-
ference of the rotors, is greater
than in a hovering regime. Conse-
quently, its required power at low
flying speeds is greater than in
the hovering regime.
Thus, a fore-and-aft heli-
copter has poor flying character-
istics at low flying speeds (in ac-
celeration, in takeoff runs when
taking off Hke an airplane, and in
towing) ; they are substantially
worse than those of single-rotor
and side-by-side helicopters.
V
Since u depends on the ratio
:, which is directly propor-
to
2.0
10 V
tional to the ratio
V
^/p~
(p being
Fig. 2. 90 Induced Velocity u vs. V for
Helicopters of Various Configurations.
the load per square meter of the
rotor area, p
-ip), the flyini
speed has a different effect on the required power for helicopters with dif- /172
ferent p. Therefore, for helicopters with a larger p, the wind in this case
lowers the required power less or increases the maximum rotor thrust in hovering.
-!!- * -x-
Thus, for calculating the torque coefficient of a lift-producing rotor and
a propulsive force with coefficients ty , t^ exposed to an air stream with di-
mensionless velocity V s 0.15, we can use the following expression:
'"/■ ^^ '"/"• ■
1.05 ty" 1
\B^ V
2^.^ =
(3.90)
=/ra^r+0.285-4^ 1.04<^V/.
182
Owing to the necessity of taking tip losses into account, the coefficients
B^ are often omitted in the term t^V so that eq.(3.90) takes the following
form: B^
OT^ = /n/v.+ 0.285^: 1;/.
(3.91)
The coefficient mp^ is determined as indicated in Subsections 3 - 6 of this
Section.
8. Determination of Angle of Atta ck and Pitch of Rotor /173
The momentum theory gives no data on the angle of attack of the rotor.
Determination of the angle of attack of the rotor a and its characteristics in a
rotor-fixed coordinate system (forces T, H), however, is necessary for calculat-
ing the rotor pitch, for refining the magnitude of parasite drag of the non- lift-
producing parts of the helicopter, and mainly for determining the equilibrium
conditions of the helicopter moment relative to the center of gravity (balancing)
and its stability.
It is obvious that, when forces with coefficients Cy and Cx are generated
during some operating regime of the rotor, the determined mean dimensionless in-
duced velocities Vy and v, over the disk must correspond to these operating con-
ditions. However, the angle of attack of the rotor may differ here and depends
on the type of rotor (hinged or rigid), on the blade shape, etc. To determine
the angle of attack of the rotor use must be made of the classical theory,
wherein the found magnitude of the angle of attack depends on the assunptions
contained in this theory.
Let us determine the angle of attack and pitch of the rotor.
To each point of the ciirves of mp^ (see Pigs. 2. 63 - 2. 74) there corresponds
a certain rotor angle of attack. The angles of attack are laid off on these
curves so that, in determining mp^ from Figs. 2. 63 - 2.74, the angle of attack of
the rotor can be located. The rotor setting is found from graphs of ty = f(a.
Bo, V) (see Figs. 2. 115 and 2.116) or from eq.(2.50) of the Glauert-Lock theory
(Sect. 3), wherein p, and X are detei^nined from eqs.(3.20), (3.2I).
If the profile power coefficient is determined from eq.(3.72) rather than
from the graphs in Figs. 2. 63 - 2.74, then a, which is the angle of attack of an
equivalent rotor (see Sect .2), is calculated from the approximate equation (3*95).
This formula is derived on the basis of the following relations:
t^ = ts.\nag-\~heCOsae^tag^k^. (3-92)
Assuming h, ~ tai , we find from eq.(3.92)
««~^-av (3.93)
183
The flapping coefficient ai^ can be e^qjressed by the approximate relation
[eq.(3.94)] derived from formulas of the Glauert-Lock theory:
«'.=2^K-£-w)-H- (3.94)
After transformation of eqs.(3.93) and (3 •94), we obtain the formula for
determining a, :
1
1—21/2
T-'Hi.-^)]- (3.95)
At large Mach numbers Mq for V s 0.4, the increment in angle of attack is
found from the graphs in Figs. 2. 85 and 2.S6.
After ffo is determined, ^J,, \, cp, and other data are found.
Having determined a, eqs.(3'l5) and (3.17) will permit finding the coeffi-
cients of thrust and longitudinal force of the rotor.
Section 4« Classical Rotor Theory. Method of ^174
Numerical Integration
When calculating the aerod3niamic characteristics of a rotor in regimes with
large V, Mq, and ty, many of the assutiptions of the Glauert-Lock theory lead to
substantial errors. For commonly used rotors, we can consider that V ^ 0.3-0.35;
Mo ^ 0.55 - 0.6; and ty close to ty , based on the condition of flow separation.
In calculating such regimes it is primarily necessary to discard the ap-
proximation of the profile characteristics stipulated in the Glauert-Lock theory:
Cy = acoQ?r and c^p = c^p , where am and Cxp are constants at all points of the
disk regardless of the angle of attack a^ and the Mach number of the blade sec-
tion.
In practice it is inpossible to give a sufficiently accurate analjrtical
expression for the dependence of Cy and c^p on a and M. Therefore, in the re-
fined calculation methods the angle of attack and the Mach number are found at
each point of the swept disk after which Cy and c^p are determined from the
graphs of the profile characteristics.
For calculating the distribution of the angles of attack, the flapping
angle of the blade p must be known; however, this can be determined only if the
thrust moment relative to the flapping hinge is known. The latter can be found
when the distribution of the angles of attack is known. Therefore, the calcula-
tion can be constructed either on the basis of determining, by the method of
successive approximations, the flapping coefficients with respect to the first
2-3 harmonics, or on the basis of determining p and -4|- by numerical integra-
184
tion of the equation of flapping; the second method of calculation, which has
become widespread, will be described below.
Practical application of such a laborious conputational process is possible
only with the use of high-speed digital computers. Under this condition, the
previously used assimptions for overcoming mathematical difficulties can be dis-
carded. Unavoidable assunptions are only those due to our lack of knowledge of
individual problems at the present state of art of rotor aerodynamics. Such as-
simptlons include:
Determination of Cy and Cxp of sections, neglecting the angles of side-
slip (equal to — =— j and variations in the boundary layer produced by
centrifugal forces arising on blade rotation; Cy and c^p of the sections
are determined from the aerodynamic characteristics of the profile ob-
tained in a plane-parallel flow.
Neglect of the effect of unsteady circulation of flow about the blade
sections, which involves a complex motion, on the aerodsmamic character-
istics of the profile.
Neglect of the fuselage and hub effects on rotor aerodynamics, and
others .
The method of calculation permits taking into account (within the assunp-
tions given above) individual features of the blade profiles and to select a
profile on the basis of quantitative
data rather than of qualitative con-
siderations, as was done previously.
The aerodynamic characteristics
can be calculated together with calcu-
lation of blade deformation and with
consideration of the induced velocity-
distribution caused by a vortex system
of arbitrary form; the conputational
effort depends on the accuracy require-
ments and on the capability of the
conputer such as memory capacity and
speed of confutation.
VCOSoC
(r-lhj,)oospH^
Vcos^cosfp
1. Formulas for Calculating Forces /17^
and Moments_ of a Rotor
Fig. 2. 91 For Detennining the Comr-
ponent of Relative Velocity of
Flow around the Blade Section.
First, let us derive formulas for
determining the conponents of the rela^
tive velocity of flow about the blade
sections. These differ from the
formulas derived in Section 2 in that they take into account the spacing of the
flapping hinges and do not consider the angles P and $ to be small.
The conponent U^ (Fig. 2. 91a, b) is directed perpendicular to the blade axis
and is located in a plane parallel to the plane of rotation (or located in the
185
plane of rotation when the flapping angle of the blade relative to the flapping
hinge is zero). As shown in Pig. 2. 91, U^ is conposed of the projection of the
fljd-ng speed, equal to V cos a sin i)!, and the peripheral speed of the section
tuC^ - K.%) cos p + Ih.h 3:
Changing to relative qualities, we obtain
(4.1)
where
t/,=^=(r-/^^,)cosp+/^^, + \/cosasin<|.=
= (^ - ^h.h) <^os p + /^ ,, + IJ. sin <)),
V COS a Tr
fi = - — =Kcosa.
(4.2)
(4.3)
strictly speaking, with consideration of flow stagnation in the region of
the rotor equal to the induced velocity y^, the flow velocity in the plane of
rotation is equal to V cos a - Vjj . Therefore, the dimensionless coefficient p,
must be determined from eq.(3.20). This in-
troduces no conplications if the calcula- /176
tion is made at a given |j,, and the dimen-
sionless flying speed V is determined from
eq.(3.20) when Vjj is already known. If the
calculation is made at a given V, then for
sinplification we will determine (j by the
approximate equation (4.3).
Longitudinal axis
Fig. 2.92 For Detennining the
Position of Blade, Hub, and
longitudinal Axis of the Heli-
copter Relative to the FUght
Direction.
and through an angle i]/ + Cq - Ps
In this Chapter, we will not discuss
blade flapping relative to the drag hinge.
The variable part of the angle of deflection
of the blade about the drag hinge is negli-
gible and it can be considered that all
blades turn about the drag hinge through an
identical angle §av = Co* Therefore, at
some azimuthal position of the blade i|;, the
rotor hub is turned through an angle ijf + Cq
toward the projection of the fljd-ng speed
to the longitudinal axis of the helicopter.
if the craft is flying with sideslip (Fig. 2.92).
The conponent U^ (see Fig. 2. 91a., c) directed along the blade axis, is
equal to
U^^^V cos a cos i)) cos p.
(4.4)
This conponent determines the angle of sideslip in flow through the blades.
186
The conponent Uy is directed perpendicular to the blade axis and is located
in the blade flapping plane (being parallel to the shaft axis when the flapping
angle of the blade is equal to zero J .
As shown in Fig. 2. 91c, the conponent Uy is conposed of the following speeds:
projection of the speeds perpendicular to the plane of rotation of the
rotor, V sin of - Vt ;
projection of the conponent of flying speed, V cos oi cos ilr ;
peripheral speed of flapping, (r - tj^.^) "^P -.
The sum of these speeds is equal to
L'^y = ( 1/ sin a — ■«) cos S — 1/ cos a cos ij) sin p — (r — / j,},) — , ( A- . 5 )
where v is the induced velocity conponent perpendicular to the plane of rota-
tion (vt in Sect .3) •
On replacing the differentiation with respect to time by a differentiation
with respect to the angle of blade rotation (i|; = uat) and changing over to rela-
tive quantities, we obtain
— i/y _ _ _
L^y = — — =(V^sina — i;)cos p — V cosacos^sin p —
(4.6)
-('■-W^ = M'-,'l')cosp-i.costl;sinp-(r-/;,.^)^.
Here the flow coefficient X(r, i; ) at nonuniform induced velocity distribution /177
is equal to
X (r, ip) == V sin a — 1) (r, 'jj) = V^sin a — ['u — A'y(r, (]>)] =
(4«7)
=.X-|-A^(r,<l-),
whence
>.= V/'sina — v,
where
(4.7')
V and Av (r, ilf ) = mean and variable portions of the dimensionless induced
velocity;
\ = average flow coefficient over the disk.
The geometric sum of the conponents U^ and Uy is equal to the relative flow
velocity through the blade section in a plane noiroal to the blade axis:
u=\^ui+ni. (4.8)
The angle P and the angular velocity ■■ P .* of the flapping motion of the
187
blade, which are detennined from the flapping equation, enter the ejqjressions
for U,, Uy, U,.
Without the sinplifjo-ng assunptions made in Section 2, the flapping equa-
tion has the form
rf2p
^H.h '"^ ^+(^Kh ^os P-'h.1,\h>^sln p =
'h.J,
or, in dimensionless fonn.
where
rf.l/2 ~V ^ hh ' «- ^•'* ^vv<«2
'Kh
0)2
QbojR*
2/
h.))
'"Kh=J f ('"-^h.h)^^-
h.h
(4.9)
(4.10)
(4.11)
To calculate the fljd-ng regimes common for a helicopter, we can assume a
small value of the angle 3 . Then the flapping equation is simplified to
(4.12)
To determine the angle of attack of the blade section a^ , we examine the
drawing (Fig. 2. 93) in a plane perpendicular to the
blade axis (view along the arrow C in Pig. 2. 91).
Figure 2.93 shows that air with a relative ve- /178
locity U, directed at an angle $ to the plane of ro-
tation, will flow over the blade section turned
through an angle cp to the plane of rotation (cp being
the blade pitch in the studied section) . The angle of
attack of the blade section is equal to
Fig. 2. 93 Speeds and
Aerodynamic Forces of
Blade Element.
a,==cp-f tarT^ — =(p+tan"' =- ■
(4.13)
(4.1!f)
(4.15)
The blade pitch in the examined section depends on the following: overall
188
pitch of the rotor 9o equal to the blade pitch in the section r = 0.7 at p =
and without cyclic change of pitch; angle of twist Acp of the section relative
to the section r = 0.7; flapping angle of the blade in the presence of a flap-
piiag compensator; cyclic change of blade pitch. The sum of these terms is equal
to
tp = eo-t-Atp — ftp — 9iSin'^ — fljcos';* — y'(y„cosw!^ 4-v„sin«'J>), (4«l6)
where
9i and Q^ = conponents of cyclic change of blade pitch, with deflection
of the automatic pitch control;
v„ and v„ = conponents of elastic twist of the blade.
The aerodynamic forces per unit length in the section r are determined by
the coefficients Cy and c^ for the profile of the section under study,- taken in
relation to a,. . Since, in detennining a^ , the induced velocity in the section
was taken into account, the coefficients Cy and Cx are taken for a profile with
infinite elongation.
The Mach and Reynolds numbers in the section are
(4.17)
M =
U
a
a
-u=
--M,U;
Ub
V
=
o^Rboj
V
bU =
abr,-, =
= "bM
V
Re=— = — "-dbU=^bM. (4.18)
Since the Re for helicopters is rather high, the coefficients Cy and c^p of
the sections will be considered (for sinplification) to depend only on the Mach
number in the section. Therefore, the aerodynamic characteristics of the pro-
file for each M are taken at Re corresponding to a given Mach and mean chord and
flight altitude:
Re^^f— ") *o.7M. (4.19)
The lift and drag per unit length of the section will then be
^_Xs^^±QU^bc. (4.20)
dr 2 ^
QU^bc^., (4-21)
dXsec.^ 1
dr
■xpi
while their conponents directed along the axes relative to the rotor, i.e., /179
the thrust dT and the resistance to rotation dQ, will read
dr dr ^ dr (4.22)
i^=^^^cos a* -^^^- Sin <».
dr dr dr (4.23)
189
Substituting, into eqs.(4«22) and (4»23), the ejqjressions for cos $ and
sin I from eqs.(4.24) and ik-25)
cosO=^; (4.24)
sin$=^ (4.25)
and the expressions for — /""' - ^^'^ — '7^'°'— from eqs.(4'20) and (4-21), we
finally obtain ^^ '^^
ar z.
or, in relative quantities.
dt
'L={cM^^c^Uy)Ub; (4.28)
dr
dr
'L={c,/J,-c,U,)Ub: (4.29)
The antitorque moment of the blade per unit length, or the section torque,
is determined from the formula (in relative quantities)
^=^^[(F-7;jcosp+7,J. (4.30)
After integrating the loads per unit length over the blade radius, we ob-
tain expressions for determining the forces and torque of the blade. Since these
quantities depend on the blade position in the plane of rotation (its azimuthal
position ijf), they are given the subscript t •
^,= f ^d7- (4.31)
'h.h
i ^^''; (4-32)
'h.h
^.=j' fK^-VJ^^^^+^ft.hl'^^" (4.33)
'h.h
'h.h
'h.h
The blade thrust is directed at an angle P to the axis of the rotor. Its
projections onto the rotor axis and onto the plane of rotation are equal to
t ^ cos p and t ^ sin g .
190
VcOSot.
On mapping the TDlade forces in /180
the plane of rotation onto the longi-
tudinal and transverse axes of the
rotor, we find the longitudinal and
transverse forces of the blade
(Big.2.94):
h^ =
^4, sin p cos iji -f ^^, sin ip;
54,= — /,), sin p sin (j) — ^4, cos (j).
(4.34)
(4.35)
Fig. 2. 94 For Determining Coeffi-
cients of Longitudinal and Trans-
verse Blade Forces.
The confjonent t* cos p creates the
longitudinal and lateral moments of
aerodynamic forces m,, and m^ . :
"^zj^^ = — ^4, cos p7j, J, cos <jj; ( 4 .36 )
'"'A+ = - ^+ cos p\^ sin if. (4 .37)
In order to determine the blade forces and moments in a dimensionless form,
i_ pa(cuR)^F and
2
the dimensionless coefficients must be multiplied by -^
-^ -i. pa(u)R)^FR.
The instantaneous value of rotor forces and moments can be found by summing
the forces and moments of all blades at a selected instant of time (one blade
being at an angle i|i> the second at an angle ijr +
+ 2 ■„ , and so on).
2tt
, the third at an angle i|r +
The average per-revolution forces and moments created by the blade are
equal to the integral with respect to ^ from eqs.(4«3l) - (4«37) divided by 2tt.
On multiplying the result by the number of blades, we find the average forces
and moments of the rotor per revolution.
In a dimensionless form, the average per-revolution forces and moments of
aerodynamic forces are determined by the e^^jressions
2ic
(4.38)
(4.39)
(4.40)
191
'"'A =i;r f '"'a /'I' = ~ 57^h.h f ^t cos p cos ^. rf^.;
b 6
2t ' 2«
/IBl
(4.41)
(4.42)
(4.43)
A force equal to the sum of the inertia forces of blade flapping is trans-
niitted through the flapping hinge to the rotor hub (Fig. 2. 95). Its projection,
directed parallel to the rotor shaft axis.
\= - J '« ^('-^hM)^''^ ^dr= -5^^0,2 ^cos p,
hJi
(4.44)
creates longitudinal and lateral moments of the rotor
27t 2i:
2ic 2ti
(4.45)
(4.46)
which should be summed with the moments of aerodynamic forces [see eqs.(4.42)
and (4.43)].
We note that the integral expressions (4.42), (4.43)* (4.45), and (4.46)
contain sin ^ or cos i|r , due to
which the moments are created by
the first harmonics of thrust and
inertia forces. Therefore, the
moments of the inertia force are
greater in magnitude than the
moments of the aerodynamic force,
since the first harmonic of t* is
Rotor axis -' f^
ju^r _. , > Since Tine iirsii narmonxc 01 t,^ 1£
df/^ ''^' small because of blade flapping.
Fig. 2. 95 Generation of Rotor Moment
Created by Inertia Forces of Flapping
Motion.
The lift and drag coeffi- /182
cients of the rotor are determined
by changing from the body-fixed
system of axes to a velocity
system:
192
/y = / COS a— A sin a; (4«47)
^^=/sina + Acosa. (4-48)
2. Method of Calculation
The initial data for calciilation are the dimensionless rotor characteris-
tics: geometric blade characteristics (variation in twist A9, relative chord t>,
and profile over the blade length), load factor of the rotor ct, aerodynamic
profile characteristics, stagger of the flapping hinges t^,^, mass and weight
characteristics of the blade (-^ = P^°'''^ .. and ^f"'" ' ), coefficient of the
flapping conpensator k.
The operating regime of the rotor is given the following dimensionless
data: angle of attack of rotor a, coefficients of velocity and lift V, ty, Mach
number M,-,, deflection of controls h, T] (or angles 9i and 83 ) .
The sequence of the calculation is as follows: In first approximation, the
magnitude and distribution of the induced velocity v and the rotor pitch 9o are
assigned. The induced velocity can be taken from eq.(3«46) or from e^qjerimental
data. The rotor pitch is determined either by the Glauert-Iock theory with con-
version by means of eq.(2.242) or is assigned arbitrarily (for exatiple, 9o = ty ) .
Let us select the azimuth with which to begin the calc\ilation fo and the
initial values of Po and Pq \fo^ brevity, we denote: p' = -g|-, P" = ^~| )•
Usually, we take ^o = or to = 270°; Po and Pq can be determined by the
Glauert-Lock theory or we can assume Po = Po = (which, for all practical
purposes, does not lengthen the calculation since the natural oscillations of
the blade decay rapidly) .
At the initial azimuth we calculate |j,, \, U^, Uy , U, cp, a^ , and M at all
radii, and then determine Cy and c^p from the graphs of the aerodynamic charac-
teristics of the profiles. Next, ~z. and m^.j, are determined, and from the
flapping equation (4.10) we find Pg . From Pg* Po» and Po we find, ty numerical
integration, P and p ' at the next azimuth and continue the calculation in this
sequence at other azimuths.
In the method of calculation conpiled and programmed by M.N.Tishchenko,
integration of the flapping equation of the blade with respect to azimuth is
performed by the Euler method with conversion. From the values of Pj, P/, P'l
at azimuth i|ri we find the preliminary values of P/+x and Pj + i at azimuth
iItj+i from the expressions
193
where A^r = ijii+i - ^^
Then, from eq.(4»10) we calculate the preliminary value of the thrust /183
moment coefficient relative to the flapping hinge (mj,,ii )i+i = f(Pi+i ,
Pi+lpr )•
Furthermore, assijming first that in the section between azimuths 1^1 and
'^i+i there is a uniformly accelerated motion with an average acceleration
-i- (Pi + Pi+i ) and secondly that Pi+ipj. can be found from eq.(4«10) with re-
spect to (m^.i, )i+i and Pi+i , the system of equations
tpr
P/+i = P/ + A4.;
P/+i = P/-| Ai^;
will yield, by the iterative method, the final values of ^(+1 and ?>i+i . Then,
knowing p/+i and Pi+i , we calculate the final values of m^.h^^j and Pj+i .
The calculations showed that, with this method, integration can be per-
formed with an interval Ai|; = 12° .
Integration of the loads per unit length over the radius, as well as forces
and moments of the blade with respect to azimuth, is acconplished by the trape-
zoidal method. For exanple.
n
1=2
1
1=1
di \ '•*-^t_,
* 2
Here, k is the niimber of blade sections (r^ = 1, ri is the root section),
and n is the n-umber of calculated azimuths-
Using the described method, we then calculate one or two revolutions of the
rotor and conpare the values of p ' and P" with those which had been at this
azimuth in the preceding revolution. The obtained value of ty is conpared with
that assigned. If these values do not agree within the stipulated acctiracy,
then the difference ty - ty is used for refining the value of 9o and
194
calciilating another revolution of the rotor.
The calculation is considered conpleted as soon as ty is equal to the as-
signed value, to the required accuracy, and as soon as P and p' in the last and
preceding revolutions coincide.
As a result of the calculation, we determine the average forces and moments
per revolution, the distribution of the section angles of attack, the thrust co-
efficient, and the blade flapping angle, which are represented as Fourier series
with an accin-acy to five terms:
5
^<^ = (+^{^„cosn^-{-t„sinn<^); (4.49)
71=1
5
p=ao— 2(«/iCos«<l> + 6„sin«^). (4.50)
Only the average induced velocity over the disk, determinable by eq.(3«46'),
was taken into account in the calculations whose results are presented below /184
in Subsections 4 - 7« The blade was considered to be absolutely rigid in bending
and torsion.
The integration interval was 12°, the number of calculated radii 12, and
the accuracy within which ty, P', and P had to coincide was JAtylnax = 0.002;
lAP'Uax = 0.002; |Ap|„ax = 0.002.
At the blade tip, ■ _ and — ^ were calculated for Cv = and c,„ corre-
dr dr ' ^
spending to Cy = 0. At sections r ^ 0.975* the calculation was made without ar^r
corrections for taking tip losses into accotmt.
The calculation time of one flying regime on a conputer performing 20,000
operations per second, is 40 - 75 sec.
3. Aerodyn^iiic Characteristics of P rofiles for Rotor Blades
Below we give the aerodynamic characteristics of NACA 230 and NACA 00 pro-
files and also of a high-speed profile suitable for use at the tip of helicopter
blades. The first two profiles are taken at a relative thickness of 12^ and the
last profile, of 9%.
The aerodynamic characteristics of the profiles were obtained from test
data on a rectangular airfoil model in a wind tunnel, with conversion to infinite
aspect ratio and to full-scale Reynolds numbers taken for each Mach by means of
eq.(4.19):
Re=f— ") 6o7M = 20-10«M.
The aerodynamic characteristics of the profiles in the angle of attack range
from -2° to 15 and Mach numbers from 0.3 to 0.9 are given in Tables 2.6 - 2.8.
195
'^XP
TABLE 2.6
PROFILE NACA 23012
Al\
—2
1.0
0.205
3.5
0.46
7
9
11
12.5
1.365
14.5
1.525
15
0.3
—0.085
0,81
1.035
1.21
1,525
0.4
-0.10
0.20
0.445
0,80
1.01
1.20
1.33
1.42
1,42
0.5
—0.085
0.225
0.485
0.85
1.0
1.185
1,24
1.25
1,245
Cy
0.6
—0.085
0.225
0,485
0.843
0.94
1.0
1,03
1.048
1.05
0.7
—0.085
0.245
0.505
0.715
0.785
0,837
0.87
0.91
0,915
0.8
—0.065
0.285
0.43
0.556
0.625
0.675
0,715
0.76
0.77
0.85
-0.065
a. 185
0.30
0.435
0.490
—
—
—
—
0.9
—0.075
0.09
0.22
0.010
0.015
0.018
0,022
0.029
0,045
—
0.3
0,008
0.008
0.05
0.4
0.008
0.008
0.010
0.015
0.023
0,0355
0.043
0.07
0.074
0.5
0.008
0.008
0.010
0.019
0.031
0.0575
0.0835
0.121
0.130
••.rp
0.6
0.008
0.009
0.0135
0,0365
0.0765
0.128
0.167
0.218
0.230
0.7
0.009
0.013
0.0275
0,09
0.138
0.181
0.213
0,254
0.262
0.8
0.0125
0.03
0,067
0.130
0.177
0.121
0.253
0.294
0.304
0.85
0.028
0.049
0.080
0.145
0.185
—
—
—
0.9
0.069
0.08
0,1075
—
—
—
—
—
TABLE 2.7
PROFILE NACA 0012
M
—2
3,5
0,3
0.4
0.5
0,6
0.7
0.8
0.85
0.9
0.3
0.4
0.5
0.6
0.7
0,8
0.85
0.9
—0,185
—0.18
—0.215
—0.215
—0.235
—0.245
—0.19
—0.08
0.0095
0.0095
0.0095
0.010
0,010
0,0245
0.0415
0.069
0.085
0.095
0,10
0,11
0.11
0,135
0,095
0,02
0.007
0.007
0.007
0.007
0.0085
0.016
0.036
0.069
0.32
0.335
0.355
0.375
0.395
0.40
0.29
0.14
7
9
11
0,645
0.835
1.02
0.665
0,85
1,035
0.71
0,915
1.08
0.75
0,91
0,94
0.735
0,81
0,84
0.57
0,65
0,72
0.50
0,61
0.71
0.40
0.56
0,70
12.5
14,5
15
0.009
0.0125
0.0165
0.021
0,009
0.0125
0.0165
0',021
0.009
0.013
0.0185
0.031
0.0105
0.021
0,039
0,074
0.0185
0.061
0,0955
0,135
0.046
0.095
0.131
0,1675
0.061
0.1065
0.141
0,180
0.0795
0.118
0,149
0.187
1.155
1.34
1,39
1.175
1.25
1.25
1.1
1.1
1.1
0.95
0.96
0.965
0.860
0.863
0.865
0.765
0.765
0,75
0.0240
0.029
0.034
0.0245
0.061
0.080
0.051
0.106
0.126
0.1095
0,171
0.186
0. 1675
0,211
0.221
0.195
0,2285
0.236
—
—
—
mi
196
TABI-E 2.8
HIGH-SPEED PB3FILE
M\
0.3
—2
-0.065
1.0
0.235
3.5
0,485
7
0.835
9
1.035
11
1.18
12.5
14.5
15
1.165
1.115
1.1
0.4
—0.065
0.23
0.485
0.835
1.035
1.10
1.09
1.06
1.05
0.5
—0,065
0.245
0.50
0.86
1.015
1.015
1.0
0.99
0.99
'>
0.6
—0.065
0.26
0.53
0.90
0.98
0.96
0.965
0.96
0.96
0,7
—0.07
0.30
0.60
0.96
0.96
0.935
0.935
0.95
0.95
0.8
—0,07
0.36
0.63
0.81
0.87
0.87
0.89
0.935
0.945
0.85
—0,12
0.325
0.55
0.77
0.86
0.86
—
—
—
0.9
0.3
-0.165
0.008
0.175
0.007
0.46
0.009
0.815
0.011
0.012
0.0245
—
—
—
0,065
0.12
0.133
0.4
0.008
0.007
0.009
0.011
0.012
0.055
0.0975
0.142
0.15
0.5
0.008
0.007
0.0095
0.0125
0.046
0.093
0.13
0.1765
0.1885
Cjp
0.6
0.008
0.007
0.010
0.025
0.060
0.110
0.1475
0.195
0.205
0.7
0.008
0.0075
0.015
0.061
0.10
0.143
0.175
0.195
0.221
0.8
0.0125
0.012
0.037
0.092
0.128
0.165
0.194
0.2125
0.2415
0.85
0.021
0.026
0.053
0.11
0.15
0.19
—
—
—
0.9
0.044
0.04
0.069
0.131
—
—
—
—
—
In the calculations whose results are given below, when M < 0.3 we took /1B6
the profile characteristics for M = 0.3, whereas when M > 0.9 the coefficients
Cy and Cjtp were determined tj linear extrapolation with respect to M = 0.85 and
M = 0.9. If the angle of attack of the blade sections varied within 72° to 180°
and -7° to -180°, the characteristics of all profiles were determined regardless
of M from Table 2.9. At angles of attack from 15° to 72° and from -2° to -7°,
a linear interpolation was made between the corresponding values of Cy and c^p .
TABLE 2.9
a"
72
0,35
1.1
105
—0.33
1,1
170
—0.62
0.04
—170
0.77
0.15
—105
0.27
1.08
—85
—0.2
1.08
—70
—7
Cy
-0.32
0.87
—0.62
0.04
FigiH-es 2.96 and 2.97 contain graphs of the coefficients Cy and c^p as a
function of or at all three Mach values.
For selecting a profile at a small portion of the blade (for exanple, at
.197
the tip portion), graphs of the aerodynamic characteristics of the profiles as
a function of angle of attack are more characteristic than the profile polars,
since the angle of attack of the examined blade section depends little on Cy of
this section and is determined mainly by the flight regime (ty, V, a) and the
blade shape. Consequently, when the profile is changed, the angle of attack of
the section is not changed (bearing in mind that ao of the profiles differ by
less than 1 to 1.5°). To select the profile for a blade as a whole, the profile
polars or the graphs of the aerodynamic characteristics of profiles constructed
as a function of o? ~ cxo are m.ore characteristic .
In Figs. 2.9s and 2.99 we have constructed the graphs of a^r and cv^^ as a
function of the Mach number (a^r is the critical angle of attack at which Cy of
the section begins to decrease or a pre-vious increase stops; Cy = Cy ^ ; a""" is
the angle of attack at which a marked increase in c^p begins, oviing to flow
separation or owing to wave drag) . Since the Mach number of the blade section
is approximately equal to
M«MoC7:t = Mo(P+Fsin\J))=rMo + Myj sini|j,
the graphs in Figs.2.9S and 2.99 give the value of a at which an increase in
profile drag begins and separation phenomena appear as a fimction of a combina-
tion of Mf 1 , Mq, r, and i|; for the blade section. These graphs will be used in
Subsection 8.
The graphs in Figs. 2. 96 and 2.97 indicate that, for M = O.3, the NACA 23012
profile has Cy^^^^ = 1.53, whereas the thinner high-speed profile has Cy^j^^^ =
= 1.18 at O'er = 11°. For the latter, a steep increase in c^p begins as soon as
a > 10.5°. The NACA 0012 profile has Cy„^^ = 1.4.
At M = 0.6 - 0.7 and at average angles of attack, the profile characteris-
tics are close together, whereas at M = 0.9 the high-speed profile is more ad-
vantageous, having the lowest value of c^p at small angles of attack and a
normal slope of dependence of Cy on a .
The results of calculating rotor profile losses for different profiles are
described in Section 3*4: At low Mq, the rotor with symmetric NACA 0012 profile
and, in certain cases, rotors with the thin high-speed profile on the blade tip
have profile losses several percentages lower than rotors with other profiles;
at high Mq, the rotor with the high-speed profile at the blade tip definitely
has the upper hand.
The maximum permissible value of the lift coefficient, in terms of flow /lA
separation (see Subsect.7) of a rotor with a NACA 23012 profile is by a factor
of 0.01 - 0.02 larger than for a rotor with a high-speed profile at the blade
tip.
The aerodynamic characteristics of profiles should include corrections to
account for the quality of manufacture and design features of the blades. The
profile drag as well as the quantity Cy^^^^ are influenced by the flexibility
and roughness of the surface (fabric skin or plywood cover, spacing of ribs),
by the presence of projecting parts especially near the nose of the profile
198
ZMZ
Cy
^
M=
0.3
/■
^
Cy
/
/
'
W
0.5
M=0.9
/
—
^/
f}
y^
'^
/
10
-4
r
-^
/
V
■ /
d
0.5
5 oc-
A
5
10
OC"
Cy
//
M=0.6
W
2
.:^
5^
^
—
=^
/
0.5
/
//
y
NRCR2W12
NflCR OOIZ-
^ *mm M^High- speed
pro fi I €
10
oC
Pig. 2. 96 lift Coefficient of Different Profiles.
Cxp
o-
■
/
~ NHLH 6o\Jfc
0.20
NnCAOOIZ
High-speed
JA^
^
/
^
/,
/
profil e
/
/
/
/
'^
_
t^'
^^'
/
^
/
f
^
/
• /
/
f
y
1
'/
/
0.10
A
/
/
J
/
<:
4
/'
y
/
f
/
/
^^ — -
/
>
*ii,^
7^
'
^
"
s=
^
:r^.
■^,
--^ —
■""^
■ ' '
f
i —
M-OJS
'M'0.3
10
15 oC
Fig. 2. 97 Profile Drag Coefficient of Different Profiles.
199
(de-icing system, rivets), leakage in the joints of the blade segments, and
local deviations from the theoretical section profile.
On the basis of calculations, it is recommended to increase the Cjjp values
of the profile, obtained from model wind-tunnel tests, by Ac^p equal to:
for blades with a nose in the form of a continuous spar of metal,
plastic, or wood and with rigid shanks: 0.0 - 0.001;
for blades of segments with metal skin and ribs: 0.0015 - 0.0025;
for blades with veneer or fabric covering: 0.0025 - 0.005.
One or another value of Ac^p is selected from the indicated interval, de-
pending on the quality of blade manuf actiore .
75
70
-
"«:
^
'
-
■^
-.
^
--
■
NACmjOIZ
NACfiOOlZ
—' — — High-speed "
profile
-
1 1 1 1 I 1 1 1 1 1
0.3
OM
0.5
0£»
«"
\
-
-
•-
-
-
\
nI
\
.^
^
\
s,
^S,
\
\
\
\
5
-
r
V
5
NKKZZOn
NKHOOU
High-speed
profi le
s
\,
^^
N
^
v
s
X
N
p
'
\
N
0.3
0.
5
-
av
Fig. 2. 98 Critical Angle of Attack of Fig. 2. 99 Angle of Attack of^- at which
Profiles as a Fiinction of M. the Profile Drag Begins to Increase,
as a Function of M.
4. Distribution of Aerodynamic Forces over the Rotor Dis k
Only the average induced velocity over the disk was taken into account in
the calculations whose results are described in Subsections 4-7; the error
thus introduced into the total _average characteristics of the rotor revolution
at large and average values of V is small. The blade was considered to be abso-
lutely rigid in bending and twisting. Calciilations show that flexural deforma-
tions have practically no effect on the average aerodynamic per-revolution
characteristics of the rotor whereas partial deformations, if the blade is in-
sufficiently rigid, do have a noticeable effect. Preassigned torsional deforma-
tions can be taken into account by substitution into eq.(4.l6)..
The calculations were performed for eight variants of geometric blade /190
characteristics, given in Table 2.10, with the following initial data: a =
Y
= 0.091; k = and 0.4; -3^ = O.9 and 1.2; I
b. h
=61 =02 =0.
In this Subsection, we will examine the distribution of aerodynamic forces
200
, #
TABLE 2.10
No. of
Blade
Variant
III
IV
VI
VII
VIII
Bl adj3 Sh ap e
Trapezoidal twisted with high-speecf
p ro f i J. e
Rectangular twisted with high-speed
pro-file
Rectangular twisted
Rectangular twisted with
symmetric profile
Trapezoidal flat with high-speea
p ro f i 1 e
Hectangular with high-speed
profile and increased twist
Expanding with high-speea
p ro f i 1 e
Rectangular with larger portion
of high-speed profile
1.0
1.0
1.0
Geometric
Twist
from
r=0 to
r=l,"deg
Bl ade Profile
From Root
to ^=0.75
1.82 0.625 2.9
1,0 1.0
1.0
1.0
1.82 0.625
1.0 1.0
0,59 1.176
1.0
1.0
1.0 7
1.0 7
1.0 7
2,9
1.0 15
0.5 7
23012
From root
to 7=0.65,
23012
From /■=0.75
to F-0.85
23012 Transitional
From_r=0,65
to r=0.75,
Transitional
From r=0.85
■to 7=1
23012 Transitional High-speed'
23012 Transitional High-speed
NACA 230.12
NACA 0012
23012 Transitional High-speed
High-speed
Transitional I High-speed
From /■=0.75
to r=1
High-speed
O
H
over the rotor disk. The asstuiption of constancy of induced velocity and the
absence of blade deformations leads to errors in calculating the forces distri-
buted over the rotor disk; consequently, the data in this Subsection are only
approximate*
Let us examine rotors vri.th blades of variants I and II in two helicopter
flying regimes: one close to horizontal flight and one close to autorotation of
the rotor; both regimes are taken at equal lift coefficients ty = 0.16, dimen-
sionless velocity V = 0.3, and Mq = 0.6. The results of rotor calculations in
these regimes are given in Table 2.11.
TABLE 2.11
ty = 0.16; V = 0.3; Mo = 0.6
Characteri sties
Horizontal Flight
Rectangul ar
Blade
a"
X
eo
tx
h
K
H
as
h
h
^1
k=0
—9.4
—0.0610
7.820
—0.0095
0.00849
0.0168
0.0997
0.0973
0.0398
0.0069
—0.0025
0.0004
0.00015
-0.0059
0.0219
0.0309
-0.0143
0.0062
0.0033
A=0.4
—9.4
—0.06103
9.957
—0.0101
0.008698
0.0162
0.09667
0.09535
0.003355
0.006043
—0.003146
—0.000548
—0.0002637
0.003116
0.0249
0.03466
—0.00963
0.0062
0,0089
Trapezoidal
Blade
k=0
-9.4
-0.0610
8.0320
-0.00795
0.00796
0.01815
0.0949
0.108
0.0408
0.0078
-0.0024
0.0005
0.00027
-0.0062
0.0233
0.0377
-0,0149
0.0078
0.0055
*=0
-10.3
-0.065
8.45
-0.01
6.0086
0.0186
0.0958
0.1096
0.0405
0.0076
-0.00276
0.000628
0.000177
-0.00623
0.0227
0.0364
-0.0167
0.00859
0.00407
Autoro tation
Rectangular TrapezoidaJ
Blade
Blade
k=0
1.4
-0.0048
3.576
0.0168
0.000475
0.0129
0.0926
0.06938
0.0367
0.00559
-0.00203
0.0003
0.00014
-0.0053'
0.0289
0.0249
-0.0120
0.00427
0.00348
k=OA
1.4
-0.0048
5.62
0.0172
0.000365
0.01327
0.09247
0.07166
0.00857
0.00515
-0.00269
0.000457
0.000175
-0.0054
0.0285
0.0267
-0.0153
0,00623
0.00317
A=0
1.4
-0.0048
3.550
0.0180
-0.00015
0.0140
0.0877
0.0772
0.0368
0.0062
-0.0012
0.0003
0.0003
-0.0058
0.0312
0.0300
-0.0138
0.0051
0.0049
202
Table 2.11 shows that, in horizontal fUght, the flapping motion of the /191
blade is greater than in autorotation and that it is greater for the trapezoidal
than for the rectangular blade. A conparison of the characteristics of rotors
with rectangular and trapezoidal blades shows that, at equal a, ty, and V, the
rotor with trapezoidal blades has larger 9o , h, ai, bi and a smaller absolute
propulsive coefficient t^; at equal tx, ty, V (see the column with o' = -10.3°)
the rotor with trapezoidal blades has a more negative angle of attack, and the
difference in the quantities Sq , h, ai , bi increases even more . This is re-
sponsible for the change in balancing characteristics of a helicopter when the
trapezoidal blades are replaced by rectangular types (for exanple, there is a
decrease in deflection of the automatic pitch .control forward, owing to a de-
crease in the longitudinal force H and in the coefficient c i) .
«f
^
\
^=0
—
N
X
W
ex
\
-tf'\
(/'=18lf
-
-
\
oc—9A°^
.\
--
<>
^5>^
-
^^^
■^
s..
r'0375'0.75 0.55 \25
(/f'90
0.25
0.55 0.75 0.975 r
0.55 0.75 0S75 r
Fig. 2. 100 Angles of Attack of Sections as a
Function of Blade Radius .
Figure 2.100 shows graphs of the variation in angles of attack of a rotor
with rectangular blades with respect to blade radius at four azimuths: 0, 90,
180, and 270° . The solid lines refer to horizontal flight and the dashed lines
to autorotation.
In horizontal flight, the angles of attack are negative at the blade root,
at t = and 270° . At azimuth t = when the flapping angle is small, the
vertical velocity conponent equal to about V(q' + 3 ) - v = V(Qr + a.^ - a.^) - v
(Fig. 2.101) is large and directed downward, due to which the angle of attack at
the blade root is small or negative at this azimuth. At azimuth i|; = 270° the
roots are close to the zone of the backwash and are washed backward and upward.
Therefore, they have large negative angles of attack.
In the middle and tip sections of the blade, the angles of attack are
203
iiiiiiim III
*^=/ffO
Fig .2. 101 Conponent of Mr Veloci-
ties Normal to Blade Axis at Azi-
muths if = and ■}/ = 180°.
greatly influenced by the peripheral ve-
locity of flapping. This increases the
angles of attack at azimuths of 270°
and 0°, where the blade is shifted down-
ward and decreases them at azimuths of
90° and 180°. The geometric twist of /192
the blade reduces the increase in angles
of attack toward the blade tip at azi-
muths of 270° and 0° and decreases them
even more at azimuths of 90° and 180° .
The angles of attack are negative at the
blade tip at azimuth of 90°.
In autorotation, the angle of attack
of the rotor is positive so that the angles of attack of the blade root sections
have a large positive value. At the blade tips, the angles of attack of the sec-
tions are less than in horizontal flight, since in autorotation the rotor has a
small pitch and less flapping motion.
The distribution of the section angles of attack over the entire rotor disk
with twisted rectangular blades is
Rectangular blade
It IJ,
= -9.4
Fig. 2. 102 Distribution of Angles of
Attack over Rotor Disk (Horizontal
Flight).
illustrated by the graph in
Fig. 2. 102 (horizontal flight). The
hatched circle in this diagram is
the zone of backwash, along whose
boundaries the section angle of
attack is close to ±90° . Regions
with negative section angles of
attack are also shown by hatching.
Figure 2.102 shows that, in
the zone bounded by azimuths of
270 - 300° and relative radii
0.7 - 1.0, the angles of attack
reach a maximal value (for un-
twisted blades, the angles of at-
tack are maximijm at r = l.O).
This region, in which flow separa-
tion_takes place on increase in ty
and V, has a noticeable effect on
rotor operation as a whole. In /193
autorotation the separation region
is located in the root portion of
the blade at azimuths of 200 - 300°.
Calculations show that the
maximum angles of attack of a trape-
zoidal blade are substantially-
larger than those of a rectangular
blade. For a rotor with a flapping conpensator, the maximum angles of attack
at ijt = 270° decrease somewhat.
204
The linear thrust is extremely unevenly distributed over the radius and azi-
muth (see Fig.2.2S), which is responsible for the occiirrence of the large vari-
able bending moments of the blade. It follows from Table 2.11 that, for the
rotor with trapezoidal blades and for the rotor with a flapping conpensator, the
variable portion of thrust increases. The rotor with a flapping conpensator has
a larger fourth haraionic. These peculiarities of rotors must be taken into ac-
count when estimating vertical vibrations of helicopters. The magnitude of the
variable thrust conponent depends on the rotor characteristic |j, (or V): the
larger \i,, the larger the variable conponent. At cos ijf - t^ , the coefficient of
the first harmonic of thrust is very small; consequently, at small spacing of
the flapping hinges the aerodynamic moment of the rotor relative to the trans-
verse axis m.
^j [eq.(4«43)] need not be taken into consideration.
0.02
0.01
-.11 '' ^
-r -^''li^
— K'O ^\ J n i\ 1 f
"\Tv l5^?^^^-
inn
^^iy30o (p°
Fig. 2. 103 Torque Coefficient of Blade
vs. Azimuth (Hoiizontal Flight).
Fig. 2. 104 Acceleration Moment in
Blade Section dQ = dXp ^
X cos $ - dY„
see.
sin $ < 0.
The torque coefficient of the blade mt
varies greatly with the azimuth
(Fig. 2. 103). At azimuths where the section angles of attack increase (quad-
rants III and IV), m^ markedly decreases. In an autorotation regime, m,., is
negative in the quadrants III and IV. This is due to the fact that, at these
azimuths, the blade sections have large positive inflow angles $, as a result of
which the projection of the lift of the blade section is directed forward and
produces an accelerating moment (Fig. 2. 104). Thus, it is obvious that, in
forward flight in autorotation regime, decelerating moments are produced in the
quadrants I and II and accelerating moments in quadrants III and I? (during
vertical descent of a helicopter in an autorotation regime, the decelerating
moments are produced by the tip sections of the blade and the accelerating
moments by the root sections).
It should be noted that a very large variable torque conponent, in a rotor
with the usual stagger of drag hinges (T^,^ < 0.05), produces a small (within
1°) flapping motion relative to the drag hinges, since the eigenfrequency of
blade oscillation is by a factor of about 4 lower than the rotor ipm, i.e., the
frequency of change of m^x.
In Section 2, we noted that, at equal ty, t^, and V, the quantity mt does /194
not depend on the anplitude of cyclic change of blade pitch, whereas a change
205
of m^ with respect to azimuth does. Actually, as shown in Fig. 2. 103, for a
rotor with a flapping conpensator m^. differs with respect to magnitude and
phase. ^
5. Aerodynamic Characteristics of Rotor
The aerodynamic characteristics of a rotor are presented in graphs in the
form of the dependence t^ = f(mt) with parameters ty, a at Mf^ = const (or V =
= const), Mq = const. These graphs are convenient for determining the coeffi-
cients tx, mt, and the angle of attack q- from values of Mfi, V, Mq, ty known
from an aerodynamic calculation.
-(7.(7/
-0.03
Fig. 2. 105 Aeroaynamic Characteristics of Rotor
(Mfi = 0.0975; V = 0.15; Mq = 0.65; a = 0.09I).
Such graphs for a rotor with rectangular twisted blades with a high-speed
profile at the tip (variant II of blades) are shown for Mq = O.65 in
Figs. 2. 105 - 2.109.
We see from the graphs that the dependence of the propulsive coefficient t,
on the torque coefficient m^ is practically rectilinear, with the exception of
near-separation values of ty at negative t^ , where the rate of increment of mt
increases owing to an increase in profile losses. In these cases, curves with
206
different ty become nonequidistant. The interval between the ciirves increases
with increasing ty, which can also be attributed to an increase in profile
losses with increasing ty .
Curves corresponding to very small values of the lift coefficient /I95
(ty < 0.1 - 0.08) closely approach or intersect the curves corresponding to
large values of ty . This means that a decrease in rotor thrust coefficient (for
e3caiiple, when using a wing on a helicopter) to ty = 0.08 and less it not recomr-
mended since, in this case, the propulsive force of the rotor does not increase.
The \:pward deflection of the curves with small ty \;pon a decrease in m,. shows
that, at small ty, the rotor is not in an autorotation regime.
"•X
O.OJ
0.02-
0.01
-001
-0.02
-0.03
Fig. 2. 106 Aerodynamic Characteristics of Rotor
(Mfi = 0.13; V = 0.2; Mo = 0.65; cr = 0.09I).
The advantages of the described graphs conprise: sinple shape of the
ciirves, facility of interpolation -upon variations in the coefficient ty, and
the possibility of using them for different solidity ratios (see Sect .6). With
the use of these graphs for calculating balancing and stability, we can deter-
mine the coefficients t and h by means of the conversion formulas (3»15) and
(3.17).
To determine rotor pitch in the calculation, we use the dependence 9o =
= f(mt) with the_parameter ty
parameter Sq at V = const, Mq
at V = const, M© = const, or ty = f(Q?) with the
= const. The graphs of the relation ty = f(a) are
207
/196
-OM
Fig. 2. 107 Aerodynamic Characteristics of Rotor
(Mfi = 0.195; V = 0.3; Mo = 0.65; ct = O.09I).
ix
H
>
"^
\,
—
S
•X
\
-
X:
:^
N
V
li m
■^
^
>
\
•v
ty = 0.06^
iy-0.08-^
ty=0.J0-^
ty-OM^
y
>
V
•ck.*
^
1
^5
%
-
S
0.0025
0.0050
1
^
-•0.0073 "1
k
>i
0.0
w
0.0125
mt\
n/i-
^
WN
X
ty=O.OB^
tx=o.ds^
/
^
^
f^
V
\
-^^
'5^
^
N
\
^
^
X
'<
*-^
0.01
o a = 5°
A 0°
• -5°
A -70°
X -75°
P -20°
^
^
sT
\
ki-'..
f
^:^<
^
'«?
^
iy-a04'
ty-B.06^
ty=OW^
>i
S
s
\
^-.
^
y
^
^
\
V
X
V
^
5»
\
^
pi\?
oa
—
?5
■
^
^
Fig. 2. 108 Aerodynamic Characteristics of Rotor
(M,i = 0.26; V = 0.4; Mo = 0.65; a = O.O9I).
208
shown in Figs. 2. 115 and 2.116.
The aerodynaniic characteristics of rotors with blades of different shapes
are not presented here since, with the accepted assunption that regardless of
blade shape the induced velocity is distributed unLfomly over the rotor disk,
the difference in the coefficient mt at given ty, t^, V, Mq is determined /197
entirely t^r the difference in nipp . Therefore, our conclusions concerning the
effect of blade shape obtained in examining graphs of mp,. in Section 3.3 remain
unchanged.
coos
'0.005
-0.010
-O.O'S
Fig. 2. 109 Aerodynamic Characteristics of Rotor
(Mfi = 0.325; V = 0.5; Mq = 0.65; o = O.091).
6. Aerodynamic^ Qha^ ^acteristics of Rotor in Autorotation Regime
The graphs of the characteristics of a rotor of the variant II in an auto-
rotation regime - polars ty = fCtj,^ ), rotor performance K^, pitch Bq^ , and angle
of attack a^ - are shown in Figs. 2. 110 - 2.113.
The graphs indicate that the rotor performance is lower than that of a wing
(for more details on rotor performance see Sect.? of Chapt.II). At small V,
autorotation of the rotor takes place at large positive angles of attack. Upon
an increase in rotor rpm (of Mq) the rotor performance drops, the pitch de-
creases, and the angle of attack increases.
It is known that autorotation of a rotor is possible in the absence of
forward speed and at any low flying speed. Therefore, the minimal permissible
speed of an autogiro or helicopter on engine failure is not determined ty flow
separation at the wing, control lability loss or spinning, as would be the case
in regular aircraft, but by the permissible vertical rate of descent. In verti-
cal descent, the rotor develops approximately the same drag as a plate (cx «
« 1.28) with an area equal to the rotor disk area, with the vertical speed of
209
7198
Fig. 2. 110 Polar of Rotor in Autorotation Regime (a = 0.091).
0.16
0.18
0.03 0.10 0.12 O.lt
Fig. 2. Ill Rotor Performance in Autorotation Regime (o = 0.091).
210
i:m.
Fig. 2.112 Rotor Pitch in Autorotation Regime.
Fig. 2. 113 Angle of Attack of Rotor in Autorotation
Regime (a = O.O9I).
211
I
descent of an autogiro being
V^
1 / =3.5 l/^ .
y 112-1.28 qq/^F y A
In autogiros, the value of p was small and their rate of vertical descent
was low.
Figure 2.104 shows that the acceleration (negative) moment in an auto- /200
rotation regime is created by the projection of lift; consequently, autorota-
tion is not possible at small rotor
[tr^lmin
0,10
-
-^-
^
-
.
-
005
m
0.20
0.30
0.W
Fig. 2. 11!)- Minimim lift Coefficient in
Autorotation Regime.
Hft. Figure 2. lift- gives a graph of
the minim.um Hft coefficient in
autorotation (ty )„!„, which is
either the autorotation limit or the
value of ty at which autorotation is
generated at very large angles of
attack and negative pitch.
In a helicopter with a large
wing, the rotor lift markedly de-
creases during autorotation, and,
since the rotor cannot have a very
small coefficient ty, autorotation occurs at a lower rpm than in helicopter
regimes .
To estim.ate the influence of the geometric blade characteristics on the
autorotation regime of a helicopter, we present the following data.
At optim-um gliding speed (V = 0.2) when Mq = 0.7 for a helicopter with a
rotor not having a high-speed profile at the blade tip, the vertical rate of
descent increases by 1.7 m/sec and the flight-path angle 9fi.p by 2°; the pitch
should be 0.5° smaller. The angle of attack increases by 1.8° while the pitch-
ing moment is retained (A«? - Aa + A9fi.p - O).
At Mq = 0.6, the deterioration in autorotation characteristics is less by
a factor of 2 - 3.
Change-over to trapezoidal blades reduces the vertical rate of descent by
0.65 m/sec and the flight-path angle by 0.8°.
7 . limit of Permissible Helicgpjter Flight_Re^ime s
"CFIow Separation limitj
As shown in Subsection l+, a rotor with flapping hinges has areas with large
angles of attack of the blade sections. In helicopter flight regimes (hori-
zontal flight, gain in altitude) these are located at the blade tip at azimuths
of 270 - 300° and in the autorotation regime, at the blade root at azimuths of
200 - 300°.
212
An increase in lift coefficient ty caiises formation of a zone of super-
critical angles of attack on the rotor.
Fiirthermore, the rotor disk contains zones of high and supercritical angles
of attack, at sites where the blade passes close to vortices shed by the pre-
ceding blades. Here the blade enters a region of high ttpward-directed local
induced velocities causing an increase in the angles of attack of individual
sections .
As soon as the zones of sipercritical angles of attack become large, the /201
rotor characteristics change noticeably: The dependence of ty on the pitch and
upward angle of attack becomes nonlinear, and the coefficients of flapping,
longitudinal and lateral forces, and profile drag of the rotor all increase.
The limit of permissible regim.es with respect to flow separation conditions
is determined by the magnitude of the rotor lift coefficient ty characterizing _
the average level of the section angles of attack, by the velocity coefficient V
characterizing the degree of nonuniformity of distribution of the section angles
of attack over the rotor disk, by the rotor angle of attack determining the
character of the distribution of the section angles of attack, and also by the
blade shape and the separation characteristics of its profile.
From the expressions for the coefficient ty and V
y G
y
1 1
it is obvious that, on a decrease in rotor rpm and an increase in flying speed
and altitude, the coefficients ty and V will increase so that the helicopter may
enter a flow- separation regime. The phenomenon associated with flow separation
at the rotor blades can be stopped rapidly by decreasing the pitch, increasing
the rotor ipm, and reducing the flying speed.
Deep penetration into the flow-separation zone sometimes ends in catastrophe
for the helicopter. One of the most inportant problems of selecting the heli-
copter parameters in designing and determining its flight characteristics is to
ensure absence of flow separation in all permitted flying regimes. Owing to the
possibility of entering the flow-separation zone, the maximum flying speeds and
altitude are limited on helicopters and any decrease in rotor rpm below an
established liaiit is inpennissible. In order to avoid flow separation at high
flying speeds, a wing is installed on helicopters to reduce rotor lift.
Flight tests show that flow separation manifests itself by an increase in
blade stresses and in blade hinge moments, increase in helicopter vibrations,
imbalance of the helicopter, and deterioration of control 1 ability . Consequently,
the manifestations of flow separation differ widely and are conplex for deter-
mining the limit of separation by calculation. Flight tests and wind-tunnel
tests of rotors yield insufficient data for establishing the overall limit of
213
flow separation. Therefore, the limiting values of the lift coefficients ty
obtained by calculation are given below.
Calculated graphs of permissible values of thrust coefficients are given
in the literatiire (Ref .20, 24). In the first of these papers, the limiting
fUght regime is that regime at which the average lift coefficient of the blade
at azimuth ij; = 270° becomes equal to the maxim-um lift coefficient of the pro-
file Cy . In the second paper, the criterion of flow separation in helicopter
flight regimes is taken as the equality of the angle of attack at the blade tip
at azimuth ijt = 270° to some critical value a^^ : Q?or ~ l2° in a regime corre-
sponding to the start of separation phenomena, and a^^ = 16° at the limiting
flight regime with a large separation zone.
A shortcoming of both methods is that one does not know how to select Cy
or Qfgj. for a blade with a set of profiles. Furthermore,, the degree of nonuni-
formity of distribution of the angles of attack over the rotor disk depends /202
on V; at ty = ty^^, and at large V, the zone of increased angles of attack occu-
pies a smaller portion of the disk than at small V. • Therefore, the appearance
of sipercritical angles of attack at large 7 has a_less pronounced influence on
the change in rotor characteristics than at small V; this is not taken into ac-
count in the method presented in the second paper (Ref .24)-
^
•
%-ir -
-
^^
^
^
^,
lc(-/l-
-_■
."
^
r-"
'^
W
0.i
^
^■
^
-
-
^^
^
--
-
00*^1
/^
y
-
-^
(
^
y
^
y'
y
^
-
'
X
X
.\
(f
^
y
0.15-
X
«o^
^
^
(
X
y
^
^
^
-
^
^
/
/
-10
-15
-10
a'
Fig. 2. 115 Change of Coefficient ty as a Function of
Angle of Attack and Pitch of Rotor
(V = 0.2; Mo = 0.7).
In the separation limits constructed below, it is assimied that the permis-
sible magnitude of the coefficient ty is the value at which the character of the
dependence of ty on the angle of attack and pitch of the rotor begins to change.
Such limits are constructed for rotors with different geometric characteristics
214
/203
Cy
1 I ,
^
g-l5 --
__ — — ^
__^ — -"^ .0 ^---'
ttZy __
u
^ ■^ " 6iiJ^^ ^
^^^
" A^-
-^r^ 'ycr -^'^
^
_
H-W|a5°
-t=^ I.^-
—
r >^
" 1/./5
/
/ /
/] [
i^jfl / •
/
/
/
/n/>
/
r
'/V
z
<?ktt/
!~ 7
>
L /I
/
/
/
" a(/J
V
'' I _ -
/
\ T /
^
/
-ZQ
'-15
-w
-5
5 cL'
Fig. 2-116 Change in Coefficient ty as a Function of
Rotor Angle of Attack and Pitch (V = 0.4; Mq = O.?)-
i
^
°f
h,\h^
^
A
-
■4
y
/
/
0.2'
o.op^
/
/
i
/
/
/
/
0.7
0.025
5.
c
o
a
/
^
^^
'--
-^
X
y
^
"
--'
/
7
1 —
-/
m
nj
L
1.
/
4—
e
o
o
1.
/
--|
777
p'*.
777
-|
/
/
/'
1
0.0
/
o.c
3
2
-
—
a
^
-
-
-
—
1"
-20 -15 -10 -5 5 oc*
-20 -15 -10 -5
5 oc°
Fig. 2. 117 Dependence of Coefficients
of Longitudinal and lateral Forces h
and s and of Flapping a^ and hi on
Rotor Angle of Attack (V = 0.4;
Mo = 0.7; eo =11°).
Fig. 2. lis Dependence of Coefficients
of Torque m^ and Profile Power lEp^ on
Rotor Angle of Attack (V = 0.4;
M„
0.7; eo = 11°).
215
on the basis of calculations lay the method presented in Subsection 2.
Figures 2.115 and 2.116 give graphs for the dependence of the coefficient ty
on the angle of attack and pitch of a rotor with blades of the variant II, at a
flapping conpensator coefficient of k = 0.4. Figiire 2.115 indicates that, at
V = 0.2 when ty reaches a certain value, the increase in ty practically stops.
The coefficient ty has a limiting value which it cannot exceed at any a and 9© .
Thus, because of the small V, the flow separation extends over a large zone and
there is a marked change in characteristics. Since the incipient deflection of
the cvccve from the linear segment is not well-defined, we will use, at V = 0.2
and for ty , a value of ty less than the maximum by an amount corresponding to
Aff = 2° (Aty « 0.01).
At V = 0, the maximum possible value of ty is taken for t^j. (Sect .8). At
large V (see Fig. 2. 116), the increase in ty with respect to a markedly slows
down at some value of ty . The value of ty at which the curve deviates from the
linear law by Ao? = 0.5° is taken for ty .
Figures 2.115 and 2.116 indicate that the quantity ty at given V and Mq
depends little on the rotor angle of attack.
Figures 2.117 and 2.118 give graphs for the dependence of the coeffi-
cients of longitudinal and lateral
forces h and s, flapping a-^ and bj,.
/204
0.35
\
s
-
-
-
\
\
S
s.
^
^
=^
s
\
V
s
\
\
-*
0.30
\
ii
X
V
\
s-
\
\
^
"^
s
\
\
<c-
>
\
\
0.25
—
-
-
\
0.20
m.i
torque m^, and profile power mpr on the
rotor angle of attack. These coeffi-
cients also change when ty = ty : The
forces of the rotor and the flapping
motion of the blades increase backward
and to the side of the advancing blade
(^If = 90°), and the profile power coeffi-
cient increases markedly. The variable
portion of rotor thrust, i.e., the
second and higher harmonics, also in-
creases substantially.
Thus, at V > 0, the value of t„
' ' 'or
is smaller than the maximum possible
values of ty ; however, it can be assumed
0.10
0.20
0,30
that, as soon as
ty =
the ■ above-
Fig. 2. 119 Dependence of ty on V
and Mq (Rectangular Twisted Rotor
with High-Speed Profile at the
Blade Tip).
mentioned phenomena associated with flow
separation will become manifest.
The_curve for the dependence of
ty on V and Mq is plotted in Fig. 2. 119*
It is obvious that the quantity ty de-
At small and medium V, tj
creases greatly ipon an increase in_V.
decreases with increasing Mq, whereas at large V
the effect of Mq is insignificant.
216
Z20i
0.35
Fig. 2 .120 Dependence of ty on V of Rotors with
Blades of Different Shapes (Mq = 0.4).
V.
\
\
N
-
1
-
\
\
n
^
\
\
1^
U-Zi
\
N
i^
\
s
'N
\
^
V
\
^
^
\
s
^
*^s
X
^
"^^
^^
^s^~
^ ^^
"^■-v^
^I
nfiT
■^/
0.10
DM
0.3
O.fV
Fig. 2. 121 Dependence of ty on V of Rotors -with
Blades of Different Shapes (Mq = 0.?).
217
Figures 2.120 and 2.121 give graphs of t„ for blades with different geo-
' c r
metric characteristics. The rotor with rectangular blades of MACA 230 profile
(variant III) has the largest value of ty . The same rotor with a highr-speed
profile at the tip (variant II) has values'" of ty smaller by 0.01 - 0.02. An
r
increase in geometric twist of the blade increases ty by approximately 0.01
(variant IV). The rotor with trapezoidal blades (variant l) has the smallest
value of t„
'or
The graphs of ty are approximate and obtained by calculation, but they
do permit the helicopter designer to determine the limit of safe flying speeds
before conducting special helicopter flight tests. Flight tests show that we
can obtain a slightly larger value of ty than the calculated values of ty
This is e^^ lai ned by the fact that our accepted ty are smaller than the maxi-
mum possible values of ty, and also by the fact that factors that increase Cy ^
were not taken into account in the calculations, namely effect of centrifugal /206
forces on the boundary layer and unsteady flow through the rotor blades.
8. Distribution of Profile Losses Over Rotor Di_sk.
Dependence of Profile Losses on Aerodynamic
Characteristics of Blade Profiles
In Section 3*3 we examined graphs of the coefficients of rotor profile
losses. Let us define the extent of influence of aerodynamic characteristics
of the blade profile, peripheral speed, and blade shape on the distribution of
profile losses over the rotor disk and their total magnitude.
The required power of the rotor, referred to all-up weight, is proportional
to mtM§ [see eq.(5.l6) in Chapt.IIl]:
^ — »<3 a , — ,,3
— = /ny.Mo , = const, /re. Mo , /■ ^-, \
G * 75iCyMl ' f (4.51)
where
/rejMo = const2-|-/re^rMn. (4.52)
Thus, the required power of a helicopter at given Mf ^ , H, p = -y-, c^ =
^°g,^ - is determined by the quantity ffippMo calc\JLated at values of CyM| and C^M^
corresponding to the given quantities. For exanple, a helicopter has a load per
square meter of the rotor disk area of p = 35 kg/m^ and a parasite drag coeffi-
cient of "cx = 0.0075; the calculated (operating) flight regime is V = 275 km/hr
at a height of H = 1000 m. Under these conditions, the dimensionless coeffi-
cients of a helicopter are equal to
CyN[l = -^-— = 0.00545;
218
/207
-JSM
Fig. 2. 122 Angle of Attack at Blade Tip Section as a
Function of M, for Three Values of Peripheral
Rotor Speed (Mq).
CrnM
300 </>'
Fig. 2. 123 Profile Losses in Blade Tip Section as a
Function of Azimuth Position of Blade, for Three
Values of Rotor Peripheral Speed (Mq ) .
219
V
Mfi =—=0.227;
CxI^l=c^N[}i =0.00039.
Let us give the resiilts of calculations pertaining to the tip section of
the blades r = 0.975. Fig. 2. 122 is a graph for the angle of attack change of the
tip section of a rectangular blade (variant II, a = 0.091) vri.th respect to azi-
muth, plotted as a function of M of the section for three values of the peri-
pheral speed (Mq). We see from the graphs that the section has large angles of
attack at small M and small negative angles at large M. By means of these
graphs, it is easy to determine the location of flow separation zones at inr-
creased profile drag. For this, it is necessary to plot the cvocres of a^^ and
0?^'" for the profile of the investigated section (see Figs. 2. 98 and 2.99) on the
graphs in Fig. 2. 122. It is obvious that, for a highr-speed profile and for Mq =
=0.7 (u)R = 235 m/sec, V = 0.325, ty = 0.1228) the maximum angles of attack are
low (1.5° lower than the critical values) but at azimuths i|f = 35 - 1^0° deep
penetration into the region of. high c^p takes place. At Mq = 0.655 (ujR =
= 220 m/sec, V = 0.347, ty = 0.14) the maximum angles of attack are close to
critical and there are two zones of high profile drag: at azimuths i|r = 55 - 120°
and at azimuths t = 270 - 0° when of = 10 - 5° and M = 0.41 - 0.62. When Mq =
= 0.61 (u)R = 205 m/sec, V = 0.373, t,
= 250 - 350° ^
profile drag. There are no increases of profile losses at large M at azimuth /208
i|r =90°.
The permissibility of deeper penetration of the tip section into the flow-
separation zone from the point of view of rotor behavior as a whole is charac-
terized by the graph in Fig. 2. 119* In conformity with this graph, a flight
regime with Mq = 0.61 is pennissible.
As shown above, the required power of a helicopter and the profile losses
of the rotor are determined ty the quantity mp^Mf which, for the examined sec-
tion, is equal to
^y = 0.l6l) the tip section at azimuths i|r =
"'° penetrates into the flow-separation zone and into the zone of high
Mo
dr 2n
(4.53)
Figure 2.123 gives graphs for the product c^pM^ plotted against azimuth.
The integral of eq.(4.53) is equal to (Table 2.12):
TABLE 2.12
Mo
0.7
0.655
0.0058
0.61
dmpr
dl ^»
0.0073
0,0056
220
Consequently, the greatest profile losses in the section under study occur
at Mq = 0.7 and the smallest losses, at Mq = 0.61. At Mq = 0.655, the profile
losses are somewhat greater than at Mq = O.6I, but local separation phenomena
are absent.
Mow let us assume that we were to change the profile in this section. Its
angles of attack would then change slightly, while the zone of flow separation
and high profile losses might change substantially. The curves of a^^ and a*
are also plotted in Fig. 2. 122 for the NACA 23012 profile. Obviously, at all Mq
the section would have no separation zones but would have a large zone of high
profile losses at azimuths i|f = 280 - - 170° . Especially high will be the
at
losses at i|( = 90°, where M is greater than M^p by 0.1 - 0.2.
all ■ii , the ipm of a rotor with this profile could be reduced.
Since a^ < a^
Thus, the graph in Fig. 2. 122 gives the optimum dependence of 0-0^ and a''
M of the profile, for the section under study at one of the design flight
regimes .
on
M,
For exaiiple, the rotor profile losses would decrease if the profile had
0.9 at Of « while retaining a""" = 7*5 - 5.5
Fig. 2. 124 Angle of Attack of Blade Tip
Section as a Function of Mach Number.
at M = 0.5 - 0.6. Then the
best rotor rpm woiild corre-
spond to Mq =0.7. The thin
symmetric high-speed profile
has a high value of M^r at
Of « 0, but a low value of o?""
at M = 0.5 - 0.6.
The profile with M„ =
= 0.8 at ex = and cv"''" =
= 14 - 7° at M = 0.4 - 0.6
woiild be suitable for the ex-
amined flight regime for the
case of Mo = 0.-61. A highly
concave profile with a small
relative thickness does have
such characteristics; however,
its use would considerably in-
crease blade torsion and con-
trols stress of the helicopter.
In selecting the profile, it must be considered that the dependence o?, =
= f(M) will be different in different flight regimes. Figure 2.124 gives a
graph for a regime corresponding to flight close to the dynamic ceiling: Mfi =
= 0.122; CyM| = 0.0103, CxM§ = -0.00012, Mq = 0.7. We see from Fig. 2. 124 that,
in this regime, the p2?ofile losses are very high. In conformity with Fig. 2. 119,
this regime lies at the boundary of flow separation. In hovering flight near /209
the ground, at a lower peripheral speed, the examined section will have a^ =
= 2.7°, M = 0.65.
By suitable selection of blade shape, a certain influence can be exerted
on the change in angles of attack at the blade tip with respect to azimuth and a
better combination can be obtained of the dependence a^ = f (M) with the profile
221
characteristics. As typical exanple. Fig. 2.125 gives a graph of a^ = f(M) for
the tip section of blades of the variants I (trapezoidal), II (rectangular),
VI (rectangular with increased twist) and 711 (eD^anding) at ty = 0.12, t,,. =
= -0.013, V = 0.4, Mo = 0.7.
Fig. 2. 125 Angle of Attack of Blade Tip Section as a
Function of Mach Mimiber, for Rotors with Blades
of Different Shapes.
Figure 2.125 shows that the trapezoidal blade, for which a reduction of
chord at the tip (b^ip < l) leads to a decrease in profile losses, has the
largest angles of attack at M = 0.4 - 0.?. The expanding blade, at these values
of M, has angles of attack by 1° lower than those of the rectangular blade. The
blade with increased twist has angles of attack by 1.7° lower at all azimuths
than those of other blade variants.
The integral in eq.(4.53) should be calculated to obtain a quantitative /210
estimate of the effect of a change of blade shape and profile.
It is clear from the foregoing that the azimuthal distribution of profile
losses in each blade section depends on flight regime, peripheral speed, and
blade profile. Main enphasis should be placed on selecting a suitable profile
in the blade tip sections, where the largest profile losses occur. For illus-
tration. Fig. 2. 126 shows the distribution of profile losses over the radius of
a blade of variant II for Mf^ = 0.227, Mq = 0.655 at four azimuths, as well as
the distribution of the average circumferential profile losses over the blade
radius. Figure 2.126 indicates that about 35^ of profile losses are accoiinted
for by the tip portion of the blade from r = 1.0 to r = 0.9.
Section 5. Vortex Theory of Rotor
1. Problems in Vortex Theoiy
The main problem in the vortex theory of a rotor lies in the determination
222
of aerodynamic loads on the blade, -with consideration of the nonuniform induced
velocity field.
The solution of this problem perniits:
1. Eefining the aerodynamic characteristics of the rotor. These refine-
ments are less inportant for the single-rotor helicopter and more inportant for
multirotor helicopters, where the
mutual induced effect is very strong
and has a substantial influence on
their fUght characteristics.
2. Determining both the constant
and variable aerodynamic loads of /2ll
the blade and from these loads calcu-
lating the oscillations of the blade
and its deformations. Without con-
sideration of the nonuniform induced
velocity field, ar^r determination of
the variable aerodynamic loads on the
blade in a mmiber of flight regimes
is quite inaccurate. Therefore, the
rotor vortex theoiy must introduce the
conponent of blade oscillations and
the determination of variable stresses
into the calculation, i.e., into the
stress analysis of the blade.
OnHy by means of the vortex theory
is it possible to explain such phe-
nomena as the marked increase in vari-
able loads on the blade and vibrations
of the helicopter in low-speed regimes
as well as the appearance of local
flow separation zones at medium and
high speeds.
Fig. 2. 126 Distribution of Profile
Losses over Blade Radius. In low-speed regimes, the induced
velocity field is particularly non-
uniform. This leads to the occurrence
of appreciable variable aerodynamic forces acting on the blade. The blades
begin to vibrate at increasing anplitude. Extensive variable stresses are set
vp in the blades. The variable forces transferred from the blades to the hub
lead to increased vibrations of the entire helicopter. The explanation of this
phenomenon is possible only by making use of the vortex theory.
At high and mediimi flying speeds, a phenomenon is observed which we can
call induced flow separation. This phenomenon is a consequence of large induced
velocities arising in the region of vortices shed from the blade tips. When the
next blades pass below these vortices, appreciable surges in aerodynamic loads
and, in certain regimes, even flow separation are created. This phenomenon was
partially described elsewhere (Ref .17; and has been confirmed in flight tests.
223
A no less iaportant problem of the vortex theory is the determination of
the induced velocity field caused ty the rotor in the stream flovd.ng past the
helicopter and its individual conponents in flight.
The character of flow past the wings of a helicopter, its fuselage, and
stabilizer is largely determined by the velocity field induced by the rotor.
The occurrence of induced velocities leads to additional downwash and to a change
in the true angles of attack of the lifting elements and hence in the forces
acting on all outer surfaces of the helicopter conponents. Therefore, to study
the flow around these parts, it is necessary to determine the induced velocities
at various points in the space surrounding the helicopter.
Thus, the vortex theory permits determining the induced downwash in the
region of the helicopter wing and its stabilizer and hence the aerodynamic
forces acting on them. Therefore, the theory also introduces the following com-
ponents into the calculations of aerodynamic characteristics: balancing of the
helicopter, characteristics of its stability, and controllability featiires in
which these forces play a substantial role.
There are other phenomena for whose calculation the vortex theory is used.
A sufficiently detailed description of all these phenomena is possible only in
special works. Therefore, in this Section we will give only a brief accoiint of
the most important elements of the vortex theory, without detailed substantia-
tions .
2. Theoretical Schemes for the Vortex Theory of a
Rotor with a Rinite Number of Blades
In the vortex theory, the rotor is replaced by a system of botind and free
vortices. This system can be represented by a vortex sheet covered with horse-
shoe vortices (see Fig. 2. 128). The /2l2
segments of these eddies located at
the blade are known as bound vortices.
Depending on the purpose of the calcu-
lation, we can use schemes in which
the blade is replaced either by a
bound lifting vortex (lifting-line
scheme), or by a bound vorticity layer
(scheme of a lifting- vortex surface).
In the latter case (Fig. 2. 127), the
blade is replaced by a system of boiind
vortices distributed over the blade
chord with some strength Yt, so that
Fig. 2. 127 Flow around Blade Profile
in Scheme of lifting-Vortex Surface.
6/2
r= Jy,^ dx.
-6/2
(5.1)
where
224
r = velocity circulation over a contour enconpassing the blade section
(Fig.2.128);
Y^o = circxolation per unit length of the bound vertices distributed over
the profile chorxi.
The scheme of a lifting-vortex surface more accurately reflects the
phQTsical pattern of flow around the blade but is more conplex in calculations.
Therefore, to sinplify calculations, the lifting-vortex surface is often re-
placed iDy a lifting- vortex line. In determining the induced velocities at a
sufficient distance from the blade, this sinplifi cation does not produce exces-
sive errors in the results and
therefore is often used in calcu-
lations. The induced velocities
close to the blade must be deter-
mined by the scheme of a lifting-
vortex surface.
During operation of the rotor,
the conditions of flow around the
blade at different radii are dis-
similar. Thus, the magnitude of
circulation of the bound vortices
varies over the blade radius . A
change in circulation is accom- /2l3
panied by the formation of so-
called longitudinal vortices (see
Fig. 2. 128). The longitudinal
vortices are a continuation of the
boiind vortices located on the
blade and form the tails of horse-
shoe vortices extending to in-
finity.
Bound vortices
V
Longi tudina
vor t ices
Transverse
vort ices
Fig. 2. 128 Diagram of Formation of a
Vortex Sheet in Circulation of Flow
about the Blade.
The strength of the longitudinal vortices should be equal to the change in
circulation of the bound vortices over the blade radius:
- —
(5.2)
where
Yio = strength of longitudinal vortices per unit length;
r = total circulation of the bouud vortices.
If the circulation of the bound vortex changes in time, also transverse
vortices will trail from the blade. The circulation of transverse vortices is
equal to the change in circulation of the bound vortices with respect to time
'tr
or
dt '
(5.3)
where Ytr is "the circulation of transverse vortices shed "by the blade in unit
time.
The strength of the transverse vortices per unit length can be determined as
1 dr
\ U dt '
(5.4)
225
where U is the velocity con^Donent of relative flow, normal to the blade axis .
Transverse vortices, just as bound vortices, form part of the horseshoe
vortices and merge along the edges with the longitudinal vortices. As a conse-
quence, the circulation of the longitudinal vortices is variable over their
length and changes by the magnitude of circulation of the transverse vortices
merging there.
Under conditions of axial flow past the rotor, the circulation in the blade
section T remains constant in time. Therefore, a vortex sheet consisting only
of longitudinal vortices will be shed by the blade. Their strength proves to be
constant over the length of the free vortex.
3. Form of Free Vortices
Under flying conditions, the free vortices shed by the blade are carried
away from the rotor at a rate equal to the relative velocity of the flow passing
through the rotor. These velocities, generally speaking, are different at dif-
ferent points of this flow. Therefore, the free vortices trailing from the
blades are carried away from the rotor at different rates . As a result, a
rather complex vortex system will exist downstream of the rotor, which, more-
over, is continuously being deformed due to the mutual interference of the vor-
tices. At some distance from the rotor, the vortex sheet begins to be dislodged
and finally loses its original form.
It is extremely difficult to take into account deformations of the system
of trailing vortices. Therefore, in theoretical methods of calculation few at-
teDpts have been made to take these deformations into consideration. Usually,
most authors assime that the free vortex sheet is carried away from the rotor /ZlU
at a constant rate equal to the mean velocity of flow through the rotor.
The con5)onents of this velocity with respect to the coordinate axes, re-
ferred to the peripheral speed of the blade tip cuR, are usually taken as equal
to |j, and Xq av (^oav being the average velocity of the flow of the stream along
the axis of the rotor, referred to cuR) .
The average flow velocity A-oav is determined by the well-known fonnula:
^oav= (^fena — ;^
With such an assunption, the trailing vortices are arranged over a downwash
spiral surface. The longitudinal free vortices are located along downwash spiral
lines, whereas the transverse vortices are arranged over the radial generatrix
of this spiral siirface. Therefore, as applied to a helicopter rotor it is
preferable to divide the trailing vortices into spiral and radial rather than
into longitudinal and transverse, as is done in the airfoil theory.
All free vortices trailing from the blades are located within an inclined
cylindrical surface resting upended on the circumference of the rotor. The
226
"vortex system enclosed ■within this siirface is usually called a vortex column or
a vortex cylinder.
Let us derive the equation of the line along which are located the free
vortices shed from the blade at an arbitraiy radius p . This Une coincides with
the wake of the blade in the flow passing through the rotor. Neglecting devia-
tion of the blade from the plane of rotation, the coordinates of this Une (see,
for exanple, the coordinates of point A in Fig. 2. 130) can be written as follows:
x= — ecosO — (a;?(i})o — "y);!
2=— QSind; \ . -
where
-" < ^ < \|fo ;
< p < R;
^Q = azimuth of the blade at the instant of time in question;
■& = azimuth of the blade at the instant of time of shedding the vortex.
All confutations given below will be based only on such a form of the free
vortices. We will disregard refinements introduced by consideration of their
deformation.
4. Determination of the Induced Velocities by the
giot-Savart Formula
If the form of the free vortices is known, the Biot-Savart formula can be
used for determining the induced velocities. This formula permits determining
— ♦
the elementary velocity dv induced at the point M by a vortex element of length
dS (Fig. 2. 129). In vectorial form, the formula can be written as
^-£%?, (5.6)
where ^
dv = vector of the elementary induced velocity caused at the point M by
a vortex element of length dS;
r = circulation of the vortex;
dS = vector of the vortex element;
t = vector proceeding from the point M where the induced velocity is /2l5
_^ calculated to the locus of the vortex element dS;
I = |t| = distance from the point M to the vortex element dS.
The direction of the vector dv is perpendicular to the plane formed by the
— * — >
vectors dS and t .
To determine the induced velocity from the total vortex, eq.(5.6) must be
integrated with respect to its length:
227
1) =
J 4it
13
(5.7)
Having taken such integrals over the length of all vortices, located both
on the blade and shed by the rotor, we can obtain the total induced velocity at
any point of space around the helicopter.
5. Use of the Biot-Savart Formula in Devej -oping the
Vortex Theory of a Rotor
Equation (5«7) can be used as basis of the rotor vortex theory. For this,
it is necessary to deterniine the induced downwash in the rotor plane and take
this into consideration when determining
the true angles of- attack of the blade
sections. After this, the loads on the
blades can be determined l^r formulas of
the type
r=.-
CyQbU\
(5.8)
Fig. 2. 129 Diagram for Calculating
Induced Velocities by the Biot-
Savart Formula.
where the value of Cy is taken with re-
spect to profile wash, for angles of at-
tack calculated with consideration of in-
duced downwash.
To determine the induced downwash
on the rotor blade it suffices to calcu-
late only the axial (parallel to the
rotor axis) conponent of the induced ve-
locity Vy . Then the downwash angle can
be approximately determined by the formula
Aa„ =
U^
(5.9)
where Ao-y is the change in angle of attack of the blade element due to- downwash.
Accoi*ding to the rules of vector analysis, the projection of the product /2l6
—4 — »
of the vectors dS x I onto the y-axis can be calculated in the following manner:
(dS X Oy = dS^l^ - dSJ^,
(5.10)
where dS^ and dS^ are projections of the vector dS, while tx ^"^ ^z ^•J'e projec-
tions of the vector t onto the x- and z-axes; the direction of the x- and z-axes
is shown in Fig. 2. 130.
228
d-+Zn
' oav
Fig. 2. 130 Diagram for Calculating Axial Components
of Induced Velocities.
Correspondingly, the axial conponent of induced velocity can be obtained
by a formula analogous to eq.(5-7):
(5.11)
Usually, the induced velocities are represented as the sum of three comr-
ponents:
'" = 'ybo -T-^sp +'yrad.
(5.12)
where
v,,o = induced velocity from bound vortices;
Vgp = induced velocity from spiral vortices;
Vpad = induced velocity from radial vortices.
Let us construct the general formulas for calculating the axial conponents
of induced velocities, without l±miting ourselves to the case y = (y being the
coordinate of the point at which the induced velocity is calculated) .
229
6. Axial Component of Induced Velocity from Bound Vortices /2l7
Boimd vortices have a circiiation equal to F,^ and are located along the
rotor blades. The subscript N denotes here the numeral of the rotor blade. We
will consider that N = 0, 1- 2, ..., z,, - 1, where z,, is the number of blades
in the rotor (see Pig.2.13o5.
The values of dS^ and dS^ entering eq.(5.1l) for the bound vortices are
equal to
^^^^'^bo ==~dQsmi/ff, J (5.13)
where ijr^ is the azimuth of the blade with the numeral M.
Substituting eqs.(5.13) into eq.(5.1l) and taking the integral of eq.(5.1l)
over the length of all rotor blades, we obtain
^=-6-' R
N=0
where
■"bo = 2 ]r^f<j,dQ, ^
4it [is ^'^ ' /3 ^^J
The values of l^ , t^ and 1 entering into K^ are determined by the formulas
ljc= r cos <^ ~ Q cos <^f/, '
4 = rsint|j — Qsin^;v.
(5.15)
where r is the radius of the circumference passing through the point at which
the induced velocity is calculated, while
'^N=% + — N. (5.16)
Here, iItq is the azimuth of the blade with the numeral W = 0.
Substituting the values of t^ and l^ into the formula for K^, we obtain
1 rsin(^ — <); )
Aa' =
4--t /•-)
7. Axial Component of Induced_Velocity from Spiral
( Longitudinal )jyortices
In determining induced velocities from spiral vortices, we must sum the
230
velocities induced by -vortices trailing from different radii of the blade. As
in the determination of velocities from bound vortices, we must define here the
total velocities induced by vortices shed from all rotor blades.
To use eq.(5.1l) for this purpose, it is convenient to divide the vortex
sheet into strips joining the vortices shed from a portion of the blade of
length dp . Then the circulation of the vortices enclosed in this strip, in ac-
cordance with eq.(5.2), is equal to — dp.
To determine the values of dS^ and dS^, we differentiate eqs.(5«5)- Then
(see Fig. 2. 130),
Using eq.(5.1l), we obtain /2l8
-" -S| I^^-^^''*'
Ar=0 »=— «
where
f.,» = . . . . .
(5.18)
^^=^T[7i-^^°^^+-&(e^^"'+t^^)]-
The values of t^ , Ij , and I entering into L^ are determined by the formulas
(see Pig. 2. 130):
/^=— ^ A//V + '^ cos tj) — Q cos S,
An
'^Oav
/^ = rsintjj — QslnO,
(5.19)
where
The value of Hn, representing the distance along a normal to the plane of
rotation, from the rotor to the point of the vortex sheet (see Fig. 2.130), can
be expressed in terms of the azimuth of the vortex sheet j? by means of the
formula
///v = ('l'o-»+^^^)U,/?. (5.20)
where N = 0, 1, 2, ..., z^ - 1.
231
8. Axial Component of Induced Velocity from
Radial (Transverse) Vortices
The circiilation of radial vortices shed from the blade in unit time is
equal to - /* • dt. In the vortex wake, these vortices are located over the
ot
radial generatrix. Therefore,
{dS,\^^= -dQ sin ^. J (5.21)
Using the same approach as above, we obtain
2 -I R
^m<i= - 2 [ [ ^ ^^"^^ "'^'
where
N=0 /=— oo
(5.22)
= — [--^sln&+-^cos»l.
4it L '3 ' /3 J
The values of l^ , l^ , and t entering into M^ are determined by eqs.(5-l9).
Integration with respect to t in eq.(5.22) can be replaced by integration
with respect to ^, bearing in mind that ?? = ilt^j + cut. Then,
z, -1 R t/v
"'«'-- 2 H ^^'"'»'"- (5.23)
A'=0 »=— 00
The functions K^, L^,, and M^ entering into eqs.(5'14), (5.18), and (5*23) /219
will henceforth be called the induction coefficients of the vortex element dS
and a point with the coordinates r, ijr, and y.
9. Integrodiff erential Equation of the Vortex Rotor Theory
To determine the aerodynamic load on the blade profile, the induced down-
wash must be determined from all trailing vortices and bound vortices of all
blades, with the exception of the blade in question, since this vortex partici-
pates in the formation of lift expressed by Joukowski's formula:
T = QUr. (5.24)
In other words, the downwash from the bound vortices on a blade with the
numeral M = must be determined by calculating the induced velocity v^o by a
formula differing from eq.(5»14) in that the term with N = is absent: ,
232
^=' « (5-25)
We then equate the lifts determined by eqs.(5.8) and (5 •24)- This jrLelds
^^-^'^ybU. (5.26)
If we limit oursel-ves to flight regimes in which it can be assumed that
Cy = c^or and U = Uj^ and if we represent the angle of attack a as
a = a>o + Aa„, (5.27)
(where §o is the inflow angle to the blade profile which woiild be present in the
absence of induced downwash Aay), and AoTy is expressed in conformity with
eq.(5-9)» then eq.(5.26) can be written in the foim
^-~^lbU^%+\clbv^. (5.28)
Substituting here the value of Vy detemiined by eqs.(5«l2), (5.14), (5 -IS),
and (5«23) and taking into account the refinement (5«25), we arrive at the in-
tegrodifferential equation analogous to the basic integrodifferential equation
of finite wing theory (Ref.28):
z. -1 R
Si
,N=\
rpjKNdQ^
N=0 »=— ~ /V=0 »=— «o
(5.29)
It is necessary to note that the functions Uj^, $o , T^ , ^ ^ , -^^ > K^,
op OJ?
Lfj, and M^ entering here represent functions of the radius and azimuth of the
blade. The function $o also depends on the flapping motion of the blade which,
in turn, is a function of aerodynamic loads and hence of the values of T^ .
Therefore, the integrodifferential equation (5.29) must be solved together /220
with the equation of the flapping motion of the blade.
It is not possible to suggest any general method of solving this equation.
In each individual case, the method of solution most siaitable to the particular
case is used in relation to the method of deterTiiining the induced velocities.
As an exanple, we mention the method presented in Section 8 of Chapter I in
Vol.11.
For a solution, the method of successive approximations is occasionally
233
used, which involves assimi i ng at first that the induced velocity Vy is constant
over the rotor disk and Tq is calculated in the first approxmation. In that
case, the terms in brackets in eq.(5.29) are calculated and the new value of Fq
is found; the procedure is continued in this manner until the solution converges.
However, it must be borne in mind that convergence of this method is en-
sured only in individual particular cases and therefore must be separately
checked each time.
Many authors, considering one or another method of successive approximations
and believing it possible to restrict the process to the first approximation,
only give a method of calculating induced velocities based on the values of
circulation F, assuming them to be prescribed [see, for exanple, Baskin and Shi-
Tstin (Ref.l6, 22)]. Therefore, it often happens that only the operation of de-
termining the induced velocities with respect to prescribed values of F is in-
troduced into the vortex theory concept.
10. Constancy of Circulation of Trailing Vortices Along
Straight lines Parallel to the Axis of the Inclined
Vortex Cylinder and Possible Simplifications
It was already noted in Subsection 2 that the circulation of spiral vor-
tices is variable over their length because they merge with the radial vortices.
Therefore, when calculating the integral in eq.(5.18T we must find the dependence
^— = f(s?). In like manner, when calculating the integral in eq.(5.23) it must
c^F
be borne in mind that — ^— varies over the" length of the vortex sheet along
with the variable «?. This circumstance coirplicates calculation of the integrals
in eqs.(5.18) and (5.23). Therefore, in calculating these it is convenient to
SF ?iT
make use of the fact that s ^ and — -^ are constant along straight lines
op ^tf
parallel to the generatrix of the vortex cylinder.
Actually, in the case of steady flow past a rotor, vortices of identical
strength will be shed from a certain radius of each blade at azimuth f . These
vortices will be carried away from the rotor along a straight line parallel to
the axis of the vortex column. Therefore, at any distance from the rotor, at a
point of the vortex sheet with azimuth j? = \|f and radius p = r, the strength of
the spiral and radial vortices is identical.
For further conputation, it is important to note that any straight line
passing within the vortex column and parallel to its axis intersects the vortex
sheet at the points
where
234
n = 0, 1, 2, ...,<»;
J? = azimuth of the rotor reckoned only in the range from to 2n;
\|;n = azimuth of the blade for which the difference {^^ - ■&) has the smallest
positive value.
It follows from the structixre of eqs.(5.18) and (5 '23) that, to determine /221
the induced velocity at some point of space, it is first necessary to integrate,
over the entire vortex sheet, a function representing the product of the strength
of an element of this vortex sheet and the induction coefficients L^ and M^,, and
then to sum the resiilts obtained from the vortex sheet of each rotor blade
separately.
However, in this case we need not integrate along the vortices conprising
the vortex sheet. At first, we can sum the products of the strength of an ele-
ment of the vortex sheet and the induction coefficients L,^ and M^ along straight
lines parallel to the axis of the vortex column. Here, by virtue of the
strength of the vortices along the straight lines being constant, this operation
reduces to summation of only the induction coefficients. Therefore, the ele-
mentary conponents of the induced velocity from these vortices can be repre-
sented as
dvs
dr^
P ="^ 2 ^N\dQM,
dv
dn
rad=-^ '^M,^\dQd^,
(5.31)
where T^ is the circulation of the bound vortex at the instant when the blade is
at azimuth f = ??.
After integrating these expressions over the entire rotor disk, we obtain
formulas for determining the axial conponents of the induced velocities in the
foi-m
\ n=0 /
\ n=0 /
rfO.
(5.32)
On the basis of these formulas, we can construct a conputational method
applicable in practice. This method was first used by M.N.Tishchenko.
It should be pointed out that, in the practical application of this method,
the volume of conputational operations is very large.
Thus, if for calculating the integrals in eq.(5.32) the circianference of
235
the rotor is divided vdth respect to azimuth into za sections and the blade into
Zr sections with respect to radius, then for calculating the field of the axial
induced velocity conponents in the rotor plane alone, the integrands in
eqs.( 5 •32) must be calculated (z^Zj.)^ times.
If we assume z^ = 72 (Aijr = 5°) and z^ = 30, then the quantity (zij,Zr)^ will
be equal to about 4«5 ^ 10^. Therefore, this method can be effectively used
only on conputers with a speed substantially greater than 20,000 operations per
second.
11. Characteristics pf^ Using the Iiftin^ Ij.ne Scheme
and Scheme of a Vortex lifting Surface
It was pointed out above that the scheme of a vortex lifting line yields
satisfactory results if the induced velocities are calculated at a sufficient
distance from the blade. However, for determining the aerodynamic loads by
eq.(5.8) it is necessary to calculate the induced velocities on the blade, /222
i.e., where the bouxid vortex is located according to the calculation scheme.
If the lifting-line scheme is used, then the induced velocities in the
calculations begin to increase on approach to the lifting vortex and vanish at
the vortex itself. This takes place in two cases:
1. If flow past the blade is unsteady and if radial (transverse) vor-
tices are formed in the vortex wake.
2. If the spiral (longitudinal) vortices shed from the blade make an
angle differing from tt/2 with the blade axis, which always takes
place in the case of oblique flow through the rotor since the blades
have slip flow.
Consequently, the induced velocities in the lifting vortex will not vanish
only in the case of axial flow past the rotor.
These difficulties can be overcome by neglecting the effect of radial voiv
tices and rotor slip. Such an approach is widespread in practice and can be
used whenever peraiissible with respect to the nature of the problem to be solved.
However, this renders the solution rather approximate, which does not always
siu-t the researcher.
The method of calculating with the scheme of a vortex lifting surface is
free from this shortcoming. Therefore, when calculating the induced velocities
in the blade region we can use methods based on replacement of the blade ty a
vortex surface, as is done in the theory of unsteady flow past an airfoil
(Ref.30). However, this renders the problem of determining the induced veloci-
ties even more conplex. Therefore, this approach is not yet in widespread
practical lose for rotor calculation, although work in this direction is in
progress (Ref.l9).
For practical purposes, we can use the method in which the free vortices
shed by the blade are divided into vortices directly adjacent to the blade and
vortices remote from the blade. After this, the induced velocities due to
236
vortices remote from the blade can be determined by the vortex lifting line
scheme while, for calculating the velocities due to vortices adjacent to the
blade, a method based on the vortex lifting surface scheme can be developed.
12. Division of Vortices into Types Close to and Remote
from the Blade; Use of "Steady-Flow Hypothesis"
To facilitate an analysis of the influence of various elements of the
vortex sheet on the magnitude of the aerodynamic load on the blade, it is con-
venient to divide free vortices into two classes (Ref.l7). The first class in-
cludes vortices immediately adjoining the trailing edge of the blade in question
and those shed from the blade during one revolution at some small azimuth angle
Aijt (Ai|; = 20 - 30°). Such vortices are called adjacent (to a given blade) and
the induced velocities caused by these vortices are called intrinsic. The
second class includes all other free vortices. These vortices are called re-
mote, and the velocities induced by them are called extrinsic induced velocities .
Such a division is based on the fact that the vortices shed from the blade
have a noticeable influence on the aerodynamic load of the blade only while they
are sufficiently close to it. Upon removing the vortices by a distance of
20 - 30° with respect to the rotor azimuth, their influence decreases but re-
sumes its former extent when the blade executes one conplete revolution and again
approaches these vortices. Thus, the blade along its path encounters not only
its own vortices but also the vortices shed from all other blades of the rotor.
All these vortices usually fall into the general group of remote vortices. /223
In some cases, it is convenient in calculations to determine separately the
velocity field induced by the remote vortices and to investigate the flow past a
blade moving in this nonuniform field. With this approach, the flow past the
blade will be similar to a flow past the wing of an airplane flying under condi-
tions of t\u?bulent air. In the same manner as for an airplane wing, when calcu-
lating the variable aerodynamic loads on a given blade, it is possible to con-
sider the effect of trailing vortices directly adjacent to the blade by the so-
called "steady-flow hypothesis". In this hypothesis, it is assumed that, in un-
steady flow past a profile, the loads acting on the profile behave as though the
flow pattern produced at a given instant of time remained imchanged for an arbi-
trarily long time. In the "steady-flow l::ypothesis", the effect of trailing vor-
tices adjacent to the blade is disregarded.
Thus, the flow pattern of the blade can be represented in the following
form: When the rotor rotates, the blade encotinters the nonuniform field of ex-
trinsic induced velocities caused by the effect of the total vortex system of
the rotor with the exception of the vortices immediately adjoining the blade.
Under the effect of this velocity field, the angles of attack of the blade sec-
tions vary constantly, and variable aerodynamic loads caused by the nonunifOrmity
of this field begin to act on the blade. The magnitude of the variable aerody-
namic loads is affected also by the free vortices adjacent to the blade and shed
from it vipon any change in the circulation flow. The effect of adjacent vor-
tices is of the same nature as that in unsteady flow past a finite airplane wing.
237
13. Instantaneous and Mean Induced yelocities and Ge neration
of Variable Aerodynanac Xoads on the Blade
Calculation of induced velocities by the theoretical scheme with a finite
number of blades makes it possible to determine the true (instantaneous) induced
velocity at any point of space near the helicopter. The true induced velocities
prove to be variable, fluctuating in time with the frequency of the vortices
passing by a given point. One can distinguish time-invariant (mean) and time-
variant conponents of induced velocities.
If we examine a point in space fixed with respect to the helicopter close
to its rotor, then the time-variant induced velocities at this point will be
caused both by bound and free vortices . However, in conparing their values, it
becomes obvious that the variable induced velocities due to bound vortices and
to free vortices adjacent to the blade are largest in value. These induced ve-
locities fluctuate with the frequency of the rotor blades passing by the point
in question. Generation of these velocities is related primarily with the forma-
tion of aerodynamic forces on the blade and is observed when the load on the
blade remains constant in time, for exanple, in hovering flight. The velocity
field induced by these vortices in this case rotates together with the blade.
Something similar happens in the case of oblique flow past the rotor in
forward flight. Here, we can distinguish a certain conponent of the velocity
field which, rotating together with the blade, does not excite variable aerody-
namic loads on it. To calculate these loads, it is necessary to define the mode
of variation of the induced velocities at a point rotating together with the /22U
blade rather than at a fixed point of the rotor disk area.
With this approach, the main generator of variable aerodynamic forces on
the blade is the nonuniformity of the extrinsic induced velocity field caused
by vortices remote from the blade. Therefore, in the first approximation we
can neglect the velocity field caused by vortices directly adjacent to the blade
and investigate only the field of extrinsic induced velocities.
14. Characteristics of the Extrin sic Induced Velocity Field
An investigation of the extrinsic induced velocity field caused only by
vortices remote from the blade permits the statement that their variable portion,
at a point in space fixed with respect to the helicopter, will be smaller the
greater the density of these vortices.
An increase in vortex density takes place, in particular, on any reduction
in forward flying speed of the helicopter. The density also increases with an
increase in rotor angle of attack, when the mean velocity of flow through the
rotor decreases and the vortex sheet is not carried away rapidly enough from the
plane swept by the blades. Such a situation specifically occurs in braking
regimes of the helicopter before passing to the hovering state. As an exanple.
Fig. 2. 131 gives a plan view of the vortex system shed only from the blade tips
in a flying regime with ij, = O.O5. The pattern shown is inconplete, since only
the three spiral vortices shed from the blade tips are given while the vortices
shed from all other blade radii are omitted. The radial vortices are also left
2?>&
Direction of flight
Pig. 2.131 View of a Vortex System
Shed by the Blade Tip in the |j, =
= 0.05 Regime.
out . However, even this pattern gives
an idea of the extremely close spacing
of vortices in low-speed regimes.
The variable portion of the ex-
trinsic induced velocities decreases
also with an increase in the number of
rotor blades. At the limit for a rotor
with an infinite number of blades, the
variable conponents of the induced ve-
locity become equal to zero.
To calculate the extrinsic induced
velocity field, it is possible to use a
scheme of a rotor with an infinite numn
ber of infinitely narrow blades. This
scheme jd-elds more accurate results, /225
the greater the density of the free
vortex system of the rotor in the
flight regime in question.
Upon changing from a given rotor
to a design with an infinite number of blades, the local effect due to vortices
innnediately adjacent to the blade is reduced so greatly that, in first approxi-
mation, it can be assumed that this scheme does not allow for the effect of ad-
jacent vortices, so that the field determined from this scheme will be closer to
the extrinsic induced velocity field, the greater the density of the free vor-
tices. Thus, a direct application of this scheme to the determination of vari-
able aerodynamic loads on a given blade is equivalent to the use of the "steady-
flow hypothesis".
The scheme with an infinite number of blades can also be used whenever it
suffices to determine only the time-average part of the induced velocity. Use
of this scheme leads to substantial sinplifications of the problem and eliminates
many difficulties arising with the scheme of a finite number of blades. Spe-
cifically, one of the advantages of this scheme is the fact that the induced ve-
locities nowhere vanish.
In detennining the downwash at the wing and stabilizer of a helicopter,
when we are usually interested only in the constant portion of the loads, the
scheme with an infinite niMber of blades can be used in all flying regimes and
yields conpletely satisfactory results. The same probably holds for determina-
tion of the mutual interference of rotors, if the designer is interested only in
their integral characteristics.
15. Vortex Theory of a Rotor with an Infinite Number of Blades
The vortex theory of a rotor with an infinite number of blades has been
quite thoroughly developed. The solution of this problem was discussed specifi-
cally in the works of G.I.Maykopar, A.I.Slutskiy, L.S.Vil'dgrube, A.M.Proskur-
yakov, V.E.Baskin, Wang Shi-Tsun, and other authors. Each of them brought this
239
theory to e-vBr gireater perfection.
While working o\it the vortex theoary of a rotor- with an infinite number of
blades, mar^ methods were suggested ■vdiioh enployed certain additional assunp-
tions-:
1. The vortex sheet trailing from a rotor is plane-. This assunption formed
the basis of the works by L.S^Vil*dgrube and L.O-.Mel»ts-, ard substantially
sinplifies the calculations.. Therefore., such an approach, wJiLoh L.S,.Fil*dgrube
brought to a form convenient for practical use, became widespread-.
2. It was proposed in a number of p^ers that the irduced velocities can be
determined rather accurately, with consideration of only the constant portion
of the circulation of bound vortices or with ttre addition of one or two first
haimonios of this circulation.
Recently,, V*K.Baskin (Eef^l6'.) and Wang Shi-Tsun ('Itef-.22) published papei^
in which they discarded these additional assunptions and brought the method of
such calculation to a state conpletely convB^nieiirt for practical use- Therefore,
an accotmt of only these two methods will be given below.
Vortex Theory of Wang Shi--Tsun_
The vortex theory of a rotor proposed by Wang Shi-Tsun has been ratter
thoroughly presented in the anxthoi^s work (Ref-SZ^^/. Therefore, only the basic
results will be repeated here-
16. Rotor Scheme
In all vortex theories examining a rotor with an infinite number a£ bUadea,-
it is proposed that the surface swept by the rotor is covered by continuously /22&
distributed 2?aidial bound vortices with circulation vardablLe ovra? its radius and
azimuth. The surtfaee swept by the rotor is corsidered to be plane- The cord-ng
angle of the blades is disregarded-
Circulation of the Dound vortices located aloing the radius in the rotor
sector with an angle A:i?-is taken as equal to
where
r = circ^alation in the blade section at the examined radius and azimuth
of the rotor;
Zt = nimiber of blades in the i*otor-
It is further assumed that t-here exists a spiral (lon^tudinal'i) free vortex
with a ciroiLlation of
d^p )-^ '^d^^» (5.33)
240
w
due to the rotor element imder study.
As a consequence of the circulation of the bound vortex, varying with any
change in the azimuth position of the blade, radial (transverse) vortices vjith
a circulation of
are also shed by the element in question.
The form of the sTorface over which the free vortices are arranged is repre-
sented as a downwash spiral surface, jxist as in the vortex theory of a rotor
with a finite number of blades . However, now the vortex cylinder is filled with
continuous free vortices rather than with discrete layers of widely spaced vor-
tices j as had been the case in the theory with a finite number of blades.
17. D etermination of Induced Velocities
To find the induced velocities due to trailing and bound vortices we used
the Biot-Savart formula [eq.(5.6)]. As noted above, to determine the total in-
duced velocities, it is necessary to sian all elementary induced velocities ob-
tained from individiml elements of all vortices conprising the vortex system.
For this^ integration in the form of eq.(5.7) must be carried out. Wang Shi-
Tsim demonstrated that there is no need here to integrate along the downwash
spiral lines along wMoh lie the trailing vortices shed by the blade. It is
sinpler to integrate along the straight lines parallel to the axis of the in-
clined vortex cylinder (AB in Fig. 2. 129), since the vortices have identical
strength along these straight Unes. Such a method was used earlier by I.O.
Mel*ts for the case of a pla,ne vortex system.
Thus_, in calculating induced velocities for the scheme of a rotor with an
infinite number of blades, it is possible to carry out integration along straight
lines parallel to the axis of the vortex cylinder rather than summation of indi-
vidual discrete quantities, as is the case in the scheme with a finite niimber of
blades (see Subsect.lO).
Using this fact, Wang Shi-Tsun was able to calculate the above integrals
and to obtain sufficiently sinple formulas for detennining all induced -velocity
conponents..
IS. Calculation Formulas for Induced Velocity Determination
The induced velocities (Ref..22) are i^resented as the sum of three types:
■"^■Vbo +^sp +'"rad' (5.35)
where _ /227
V = total induced velocity;
v^,, = induced velocity due to bound vortices;
241
_vbp
induced velocity due to spiral (longitudinal) free vortices;
Vpad ~ induced velocity due to radial (transverse) free vortices.
All induced velocity conponents, entering eq.(5.35)» pertain to the peri-
pheral speed of the blade tip cuR.
Of greatest interest for practical application are the axial conponents
(parallel to the rotor axis) of the induced velocity. In setting \sp the problem
of calculating the rotor blade loads, axial induced velocity conponents need be
determined only in its plane.
We will give the calculation fonnulas for determining the axial conponents
in the rotor plane only. The formulas for other conponents, determinable out-
side this plane, can be found elsewhere (Ref.22).
The axial induced velocity conponents due to bound vortices are determined
by the fonnula
I 2it _
(5.36)
where T is the circulation in the blade section referred to cuR^ :
(5.37)
The axial induced velocity conponents due to spiral and radial vortices are
determined by the formiilas
Here,
where
V.
'sp
__ffc_
X
8n2 Y^ + x=
12b _
X j" r|£f(|x-f esin&yi-Qcos&/2ld»a:e,
12r.
■"r«^=
_ ^b
8ll2
/(^^ + ^L
X
m^
cos &/j -]-sin ^/j] d^dq.
/,-
(C sin 0— r sin <1/) j/^|ji2 + X
2
oav
^2 =
/2 |/(l2 + ^oav +f^' (e COS » - r COS i/)
(- Q COS » + Fcos ^) ]/"fl^ + X^^^-Tft
/2]/ ^2 + X2^+ p./ [q cos » - r COS ^ ] '
/"= Yq"^ +^^ — 2q r cos(& — <jO.
(5.38)
(5.39)
(5.40)
242
For the variables entering eqs.(5»36) and (.5'3S) we will use the following
notations :
p" = relative radius of the blade section shedding the vortex;
r = relative radius of the rotor at which the induced velocity is de- /22S
termined;
2? = azimuth of the rotor shedding the vortex;
i|r = azimuth at which the induced velocity is detemiined.
Equations (.5'3^)f (5»36), i5'3&), and (5*39) permit determining the induced
velocities if the circulation T(^, p ) is known at all azimuths «? and radii of
the blade.
n
All
%
i
■-^' 2 3
^
8 9
%
%\
10 II
Fig. 2. 132 Distribution of Circula-
tion over Blade Length, Used in the
Calculation.
It should be noted that, in the
vortex theory of a rotor with an infinite
number of blades, one usually determines
the total induced velocities, including
the conponents due to bound vortices
participating in the production of blade
lift.
19. Applicat ion and E valiiation of the
Possibilities of the Wang Shi-Tsun
Vortex Theory
In the practical application of the
Wang Shi-Tsun vortex theoiy, just as in
a number of other schemes, the distribu-
tion of circulation V over the blade length is usually represented as a stepped
line, as shown in Fig. 2- 132. The quantity ^_ ' dp is taken as approximately
equal to
Bp
dr
dQ-=r,-r,_^^i:,rQ,
{5. hi)
where AF^ is the difference of circulation at two adjacent portions of the blade.
Such an approach is equivalent to replacing the vortex sheet by a number of
discrete vortices. As a consequence, the induced velocity approaches infinity
at points where these discrete vortices are shed by the blade. To avoid this,
the induced velocities should be calculated with respect to the midsection at
constant circulation.
In determining the circulation derivative with respect to blade azimuth,
the circumference of the rotor is also divided into a finite number of sections
and the derivative
da? is taken as approximately equal to
^d^-r,-^,_,^^^,:
<.5.k2)
2k3
Thus, for a nimierical determination of the integrals entering eqs.(5«36)
and {5'3s)f we should first determine the values of Fq, AF^, and AF^ , whose num.-
ber N -will be equal to
where
//=:
(5.43)
Zr = ntmber of sections over the radius into which the blade is divided;
Aijf = pitch with respect to blade azimuth over whose length the circula-
tion is considered constant.
If we take z = 30 and A^f = 5°» a determination of the induced velocities
will require 2l60 values of Fp, AFp, AF^ (a total of 64S0 values). To determine
the induced velocities at all 2l60 points of the rotor, the operations needed /229
for determining the integrands in eqs.(5.36), and (5.38) would have to be re-
peated (2160)^ tim.es. It is obvious that the time required for this would handi-
cap the practical applicability of this method, even when using high-speed com-
puters. Therefore, the practical application of the Wang Shi-Tsun theory in
such a form is possible only for a
rather limited number of sections,
taken over the radius and azimuth of
the blade. On high-speed conputers
with a rate of 20,000 operations per
second, the calculation can be per-
formed under approximately the follow-
ing conditions:
= 10 - 12°.
Zr = 10 - 12; At =
Fig. 2. 133 Diagram of Formation of
Vortex Rings.
This shortcoming is eliminated if
the circulation F and induced veloci-
ties are e^qjanded in Fourier series in
harmonics and if the process is limited
to considering only a certain number
of lower harmonics necessary for per-
fonning the calciilation with satis-
factory accuracy. For practical pur-
poses, it usually suffices to restrict
the process to the first 6-8 har-
monics. This method was used in the
theory of V.E.Baskin (Ref.l6).
Vortex Theory, of V.E.Baski.n
The theory of V.E.Baskin was circiolated in a limited edition in 1955 and
later presented by the author in an urpublished report (Ref .16); as a conse-
quence, the theory of V.E.Baskin is not very familiar to specialists interested
in this problem. For this reason, the theory will be presented here in consider-
able detail but with certain sinplifications which the author did not make.
2hk
2Q. Sc heme of Rotor Elxgw
The vortex system, shown in Big^2.'l2S and. consisting of bound and free
spiral and radial vortices, can be represented in a somewhat different form.
We can consider that, during operation of the rotor, the blade continuously
sheds infinitely small closed elementary vortex rings, with a circulation con-
stant over the contour of the ring etjial to the circulation of the bound vortex
r in the blade section at the instant at which the vortex ring separates from
it ('Fig.^2..13S0-.
Since the circtilation of the bound vortex varies over the radius and azi-
muth of the blade, the vortex riiags trailing frean it will have a different
circulation at different points of the vortex sheet. Consequently, . the free
vortex sheet can be considered as consisting of continuously distributed, in-
finitely small vortex ri-ngs with different circulation T .
It is known from hydrodynamics Usee (iRef.^l, p..-266.).'] that the induced ve-
locities due to the vortex ring contour with a
^i circulation T will be the same as those due to
the layer of dipoles covering the surface
stretched over this contour, with an intensity
<ojQ^ per unit area of
230
B — r
(5.44)
Fig.-2.134 For Determining
Induced Velocities from a
Dipole Goliann.
and oriented along a normal to this surface.
Here, the signs are selected on the basis that
the circulation flow producing blade lift and
the dipoles whose vector gives a positive projec-
tion onto the y— axis, are considered positive
(iFig.-2.134X.
Consequently, the free vortex sheet trailing
from the blades can be replaced by a surface
covered by a dipole layer.
On changing to a rotor with an infinite
number of blades, this equality is somewhat modi-
fied. Aft of a rotor with an infinite number of blades, the vortex sheet fills
the entire volume bounded by the downwash cylindrical surface tangent to the
rotor circumference. This vortex system, as already mentioned, is called a vor-
tex column. This column can be represented as fill fled with dipoles rather than
with vojrtices. To determine the intensity of dipole distribution in the col\min,
a layer of height dH, filled with vortices during the time dt, is cut from the
coltimn. During this time, vortex rings with a contour boimding the area
dF==UattiQ
(5.45)
trail from all radia_.
This contour can be replaced by dipoles with an intensity D = -F . On
changing to an infinite number of blades, these dipoles are distributed over the
245
entire rotor circimiference of radius r with an intensity of
^=-|7^''^- (5.46)
If the circulation T is ej^jressed in terms of the aerodynamic load per unit
length from eq.(5«24), then
D=-£iI^. (5.47)
2iter
We can determine the quantity dt entering this expression as
UN
dt=-
(5.48)
where Vy is the average velocity of flow through the rotor:
Let us then introduce the concept of relative aerodynamic load, so that /231
T=^ciebo,^WP. (5.49)
Then,
4 " '■ ^y
Thus, the problem reduces to determining the induced velocity field caused
by the downwash column of dipoles whose intensity in a layer of thickness dH is
determined by eq.(5.50).
21. Determination of Induced Velocities from
the Dipole Column
Figure 2.134 shows that the plane of each of the elementary vortex rings
trailing from the blade is inclined at some angle i to the plane of rotation of
the rotor:
o«^. (5.51)
Correspondingly, at this same angle to the plane of rotation are inclined
also the axes of the dipoles which, in V.E.Baskln's scheme, replace the ele-
mentary vortex rings.
Henceforth we will disregard this angle and will assume that the axes of
all dipoles are directed perpendiciilar to the rotor plane. Baskin (Ref .16) did
not use this slnplificatlon in deriving the basic relationships, but it is
shown that - in calculating the axial induced velocity conponent - this is per-
missible .
246
To determine the induced velocities, let us examine a flow caused by di-
poles filling an element of the vortex cylinder of a height equal to dH (see
Fig. 2.136).
Summing the induced velocities caused by all elements of the vortex column,
we can obtain the total induced velocity.
22. Fluid Flow Induced bv a Disk Covered with Dipoles
Fluid flow induced by a disk covered with dipoles can be determined as the
solution of the laplace equation with certain boundary conditions, which will
be discussed below.
u .
The laplace equation in cylindrical coordinates
(Fig. 2. 135) can be written in the form
duZ "■" (Jq2 ""^ e dQ q2 (JS2 ~" '
(5.52)
Fig. 2. 135 For De-
tenninlng the Ve-
locity Potential by
the laplace Equation.
where 9 is the velocity potential.
The velocity potential can be ejqsanded in a
Fourier series. Then,
= 2 ('PmCos/ra»+ cp„SinOT»).
(5.53)
m=0
Substituting eq.(5.53) into eq.(5.52), we obtain the equation for determining
the coefficients of the sines and cosines entering eq.(5.53)!
Solution of eq.(5.54) will be sought in the form
9« = e^*''5C(0).
Substituting eq.(5.55) Into eq.(5.54), we obtain
x"(Q)+-^x'(G)+(^^-^)z(0)=0.
(5.54)
/232
(5.55)
(5.56)
This equation can be reduced to a canonical form, if we set
Aq=q.
Then,
247
The general solution of this equation can be .vretitten in the fonn
The ftmction I^Cp^) tends to infinity as p" -» Q. Therefore, if we restrict
ourselves to a finite value of the potential, then
<P,„=^e±*".y„(*Q). (5,.59)
If we use the condition that, as u -♦ <= also cpo " 0» then the solution of
eq.(5«54) can be written as
<P,„=yie-*«y„(,*Q), (5.60)
where k 2: Q.
The function cp„ will be the solution to eq.(5-.5'^) at any value of k = kj,.
Consequently, the solution of eq.(5-«54) will also be the sum
<P.„.=2^i«~*'"^(^v.Q) ^5-61)
and the integral
'P.nr^JA{k)e-'»'J„(jkQ)dft, ('5.62)
where A(k;) is an arbitrary function detennined from the boiindary conditiona.
23.. Bourdaaw Gonditions
It is known from hydrodynafflics tsee (iEef-.-31, p..-82)j] that the velocity po-
tential on both sides of a surfaoe covered ty dipoles is associated with the
density of these dipoles by the relation
il-=<P2'— <pi, (5-. 65)
vheve
D = density of dipoles per unit surface, with the positive direc-
tion of the dipole asd-s coinciding with the u-axis-;
91 andcpg = velocity potentials on both sides of a surface covered by
dipoles (iEig.-2.136:)-.
The fluid flow on both sides of the disk covered by dipoles is subject to
equal conditiona. Therefore, we can set
fjt=— <pi=(p. (5.64)
248
Fig. 2. 136 Diagram of Rotor Flow Used in
the Calculation for Determination of
Induced Velocities.
Then the value of the po-
tential at the boundary of the
disk on the side where u > 0, can
be defined as
9=-fD.
(5.65)
The boiindaiy conditions for /233
flow induced by a disk covered by
dipoles of a density D are deter-
mined from eq.(5.65) if p ^ R. If
p > R, the potential in the disk
plane is everywhere equal to zero
(cp = 0),.
The intensity of the dipoles
covering the disk can be repre-
sented in form of a serieg::
-^j9rt(B)sIn/rt6].
('5-.^6.)
Then, using the boundary condition ('.5-.65-) for detewQining the ai*bitrary
function A(k) in eq.(5«62), we can write the Solution of the (jontiniiity equa-
tion (5 '52) in the form
9=-- ^2]J'^''*"-^'«(*e)*x
m=0
The derivation of this ejqjression is given elsewhere (Ref.3l)-.
X
(5.67)
24. Transform ation of Eg. (5.6?) t o th e Rotor Axes; Use of
the Theo rem of Addi tion of _gylindricai Functions
Equation (5.6?) is written in cylindrical coordinates with the u-axLs going
through the center of the disk of thickness dH, cut out from the dipole column
(see Fig. 2. 136). To transfonn this ejqpression into coordinates related with the
rotor, we can use the theorem of addition of cylindrical functions (Ref .34). /234
It follows from this theorem that
cos/n»y„(Ac)=(-l)« 2 Jnikr)Jn-^m{kl)cosn<^, I
(5.68)
249
sinm^J„(kQ)=-{-^iy" 2 J„ikr)J„+„{kl)sinn'^.
The velocity potential from the dipole layer is ejipanded in a Fourier
series:
?= '^i'?nCOsn<^-\-<f„sinn<^).
(5.69)
Then, using eqs.(5«68) and (5 '69) and equating the coefficients of cos nj(f
and sin raji with identical n in eq.(5.67), we obtain
<?«=
■2( _ 1 )m j e-*„ y^ (^kr) y„^„ (kl) kdkX
m—O
XJ5„(e)-/m(*Q)Q"fQ,
)
^(-ir ^ e-'"'J„(kr)J„+Jkl)kdkX
m=0
R ^
X^D^(Q)J„(kQ)QdQ.
IS
(5.70)
25. Determination of the Total Velocity Potential from
the Entire Dipole Column
To determine the conponents of the total velocity potential represented as
^= '^(^nf^osn.'^ + ^^sinn'^), (5.71)
eqs.(5.70) must be integrated over the entire dipole column (see Fig. 2. 136).
For this, we must first write out the values of D„(p) and D„(p).
We note that within the vortex column the intensity of the dipoles, just as
the strength of the vortices, is constant along straight lines parallel to the
column axis. Therefore, the dipole intensity D(p , ^) in arQr layer cut out from
the vortex colimm will be equal to the intensity of the dipoles trailing from
the rotor at a point where r = p and ^ = ■&.
If the relative aerodynamic load P [see eq.(5.50)] is represented in the /235
form
(5.72)
250
then we can write
4 " Vy Q
4 " Vy Q
Setting
^=^7^1 (5.74)
and substituting eqs.(5»73) into (5.70), we can integrate the obtained expres-
sions with respect to H. It is easy to demonstrate that, for perforaiing this
operation, we must determine the value of the integral
JiH) = °°^e-''^J„^„(^k'^H)dH. (5.75)
Referring to the handbook by Ruzhik [Ref .34, p. 721, eq. (6. 611.1)], we obtain
JiH)=±^A,TP, (5.76)
where
"""^yTT^: (5.77)
■"- ,. ■• (5.78)
Here p = m + n, and eq.(5.76) is valid only if p > 0. Therefore, at m +
+ n < 0, eq.(5.76) takes the form
M/^)~^A,{~\y'cP, (5.79)
where p = -(m + n).
Let us introduce the new variable
z=kR. (5.80)
Then, using eqs.(5.76) and (5. 80), we can write out the expressions for /236
the conponents of the velocity potential from the entire dlpole column:
251
X ] e-^~"J,{z?)dz \P„(Q)J„izQ) dQ,
m=o
X J e-^~"J,iz7) dzlp„{Q)J^(zQ)dQ.
(5.81)
Here, all linear dimensions pertain to the radius of the rotor R, including
y(? = ^).
26. Determination of^ Inducted Velocitie s
As already mentioned, we will determine only the ajcLal induced velocity
conponents. For tMs., we must take the derivative of the velocity potential
with respect to y. We see from eq.(5.81,) that only the term e~^^ depends on y,.
Therefore, the operation of differentiation leads to e:^ressions differing from
eq.(5«81) only in sign and in conppnent z.
Before writing out the final formulas for determining the induced veloci-
ties, we will present them as a Fourier series:
^= 2'(^n cos «++X„ sin «<];).
(5.82)
Here, all induced velocity conponents are referred to the tip speed of the
blade (uR. In determining the induced velocity conponents written in form of
eq.(5.82), the following operations must be performed:
" <^R\dy dy )'
)•
dy
" u/? V dy
dy
{5'&3)
An exception is the determination of Xq, which is calculated as
r- 1 _ ^«g"o
0)/? dy
(5.84)
As a result of the actions provided for in eqs.(5.83) and (5.84), we obtain
252
o'
M
m=0
X ] e-'~«j„{£'r)zdz J ^„(g)y^(zo)de.
Z222
(5.85)
('5-86)
It is assijmed in these expressions that n > m. Therefore, in conformity
with eq.(5.79)^ when n < m, in place of t"""" we must calculate (_ii)°'-°j»-° .
When performing operations with eqs-.(5-«83) we must bear in mind that
y_„(£:7)=(-i)''y„(z7.).
Different methods can be s'uggested for calculating the irttegrals entering
eqs.(5»85) and (5^86)^ One such method will be given in Section 8 of Chapter I
of Vol.. II, where a method of calculating elastic vi'brations of a blade with con-
sideration of a variable induced velocity field will be disxjusised-.
It should be pointed out that, in certain particular cases, the integrals
eriterirg eqs-.('5-.85^ aiid ('5-* 86) can be calculated analytically (see Sect -8-. 6 of
Ohapt-I in Vol- II) ^
The vortex theory af V-E-Baslcin permits obtaining results rather rapidly,
if the person doing the calculations has at his disposal a ccoputer with a rafte
of at least 20,000 qperations p&r second- At this rate, the calculation cain be
penformed in about ^ min. Therefore, if the problem is to determine the mean
induced velocities with an accuracy to high harmonics and if the assunption that
the vortex sheerfc is plarne cannot be used, the best method for a determination Of
induced velocities is the Baskin vortex theoiy..
Section 6- Bsgaerimental Determination of Aerodynamic
Qharaeteristios of a Rotor
The most reliable methods of determining the aerodynamic characteristics of
a rotor are flight tests of the helicopter with the rotor under study or with
another similar rotor wlrdeh can be regarded as a model of the investigated rotor,
as well as wind-tunnel tests on full-scale rotors or large-scale models.
253
In this Section, we -will present certain results of flight tests and wind-
tunnel tests and give a brief description of the testing procedure. The tests /238
were carried out at research institutes ty M.K.Speranskiy, A.I.Akmov, and
others .
1. Flight Tests for Determining the Aerodynamic
Characteristics of a Helicopter
The aerodynamic characteristics of a helicopter in a system of wind axes
(Figs. 2. 105 - 2.109), i.e., in the form in which they are used in aerodynamic
design, can be obtained from flight tests.
Selecting the flying speed, rotor rpm, flying weight, and flight altitude,
flight tests will furnish constant values of M^j^ , Mq, and the thrust coefficient
of the helicopter ty (below, p and T denote the pressure and tenperature of the
air) :
V
f^fl = — = 7r= = const, — ?^;
^ a 20.1 /r ' -/t '
Mo=- = const2-^^;
a y T
since ±- Qa= = -^ (o.379 ^) (2Q.1 Yff=com{,p.
Fulfilling, under these conditions, various flight regimes with different
engine powers - gliding at a different rate of descent, forward flight, and
climbing with a different rate of ascent, we can obtain a sufficient number of
points of the aerodynamic characteristics of the helicopter in the form of the
dependence t^ = f (nit ) for constant values of V, ty^ , and Mq . In an autorotation
regime of the rotor, we obtain a point with m^ « (Fig. 2. 137). The point t^
= corresponds to a forward flight regime since, at this point, the propulsive
force of the rotor is balanced by the parasite drag of the helicopter. Gliding
of the helicopter corresponds to intennediate regimes. The regimes t^j^ < 0,
i.e., a gain in altitude at a point where the engine power is maximal (mt =
= mt ), are obtained from the regime of maximum rate of ascent for a given
flying speed.
As a result of tests, we will determine the aerodynamic characteristics of
a helicopter which differ from the rotor characteristics in that the aerodynamics
of the no- lift-producing parts of the helicopter is taken into account.
Thus, flight tests for determining the aerodynamic characteristics of a
helicopter involve "flying by the seat of the pants" in which gliding, gain in
altitude, and forward flight are performed at constant values of V//T, n//Y, -^.
It
254
Hfi ''Const
^
Autorotation VjCOHSt
\ ty=const
t
\ Gl irfing
\ „it"ei
\ 1 flight
\ ""i
Gain in altitude—^ \
In addition to these quantities, the follow-
ing are measured: vertical velocity Vy ; in-
clination of the plane of rotation of the
rotor to the horizon (pitch angle of the
helicopter) ■&; conponents of cyclic change
of pitch cpi, ^ (or angles of deflection of
the automatic pitch control h, T] and the
flapping coefficients); setting angle of the
blade 9o ; and torque of the rotor M,. . The
torque is measiored iDy strain gages mounted
to the rotor shaft or to the rod of the re-
duction-gear frame.
The aerodynamic characteristics of a /239
helicopter are determined from the e^^jres-
sions derived in Section 1 of Chapter III:
G cos 6ji.p
1 1
(6.1)
Fig. 2. 137 Coefficients of
Forces and Torque of Rotor in
Various Flight Regimes.
/«^ =
Mf
^=
Q(u>R)2RaF
G sin Qfi,p
1
-,Q((»R)2aF
(6.2)
(6.3)
The flight-path angle to the horizontal is
Ofifi = Si'n
-/^v
(6.4)
-Equivalent plane
of rotat ion
The rotor angle of attack and
the equivalent angle of attack are
found from the following formulas
(Fig. 2. 138):
a^%-QfLp; (6.5)
ae = a — (p,=a-|-£^i* — OjTi-l-Afe,, (6.6)
where
DiK - DgTl = deflection of the
equivalent plane
of rotation of
the rotor from
the design plane
of rotation at inclined automatic pitch control;
kbi = additional deflection of the equivalent plane of rotation of the rotor
Fig. 2. 138 For Determining the Angle of
Attack a and Equivalent Angle of Attack
Q-g in Flight.
255
B
in the presence of a flapping ccaipensator.
At a methodologically correct conduction of tests, the root-mean-square
error in determining ty is 0.5^ while, in determining mt, it is 3«5^«
The results of flight tests with the helicqjter Mi-1 are plotted in
Fig. 2. 139. The helicopter had a three-blade rotor with trapezoidal twisted
blades, D = 1!(..3 m, a = 0.0504, blades with plj^ood planking except for the /240
blade root (r < 0.59) where the shank portion of the profile was covered with
fabric .
-m -^
-0,02
-0.03
-OM
III 1 ^ 1 L
,ot=6,Z' ^Mfi =0.1 --
V =0.20
("-^ M,=O.S ~^
^
St ^
a5 l^lt
A^
XA
S^
^^
-^^^ J^
^X
^tt.
X S^
^- -2-5°A^
U
SA
- -zr^^V/Z',-
aoos'\\\ 0,010 ml
-^WU-L. _j
v\ >
\\\h-0J7
tr0.n\\
\W,^-rOJi
V\'\ 1
\V ^ fCI
-75,-l:\a;^
15...-^,
-I^S'W
-+- '^-^-v^
:5tK
^
±
Fig. 2. 139 Coefficient of Prqpulsive Forces as a Function of
Coefficient of Torque (Flight Tests of Mi-1 Helicopter).
The aerodynamic characteristics of the rotor are obtained by subtracting
the coefficients of forces created by the nonlifting elements from the coeffi-
cients of forces of the helicopter. Ely analogy with eq.(l.3) from Chapter III,
the coefficients of forces of the rotor are equal to
256
- v^
- V^
^x = ^x. - ^Xf.^r= ''A ~^'~'
(6.7)
(6.8)
where c
Cy^S,
yf
and Cj =
Sc^S
The coefficients Cy and c^ of the Mi-1 helicopter were determined hy test-
ing a fiill-scale fuselage in a wind tunnel. The lift coefficient of the fuse-
lage 'Cy is very small and we can disregard it in eq.(6.7).
It is necessary to bear in mind that the aerodynamic characteristics of /2hl
a rotor obtained from flight tests take into account the mutual interference of
the lifting and nonlifting elements of the helicopter, a fact that increases the
value of these data.
2. Wind-T'unnel T ests for Determining the Aerodynamic
Charac teristics of a Rotor
To determine the aerodynamic characteristics of full-scale helicopter
rotors in a wind tunnel, the TsAGI has special facilities for testing two-,
three-, and four-blade rotors with a diameter vp to 15*5 m.
Fig.2.15t.O Facility for Testing Full-Scale Rotors
in a Wind Tunnel.
The first facility for testing full-scale rotors with an engine power of
575 ip was created imder the sipervision of M.L.Mil' on the basis of the Mi-1
helicopter. Figure 2.1^0 gives a view of the Tiiiit mounted to the vpper structure
of the tunnel balance. The unit has a special danping suspension support, main-
taining the permissible level of vibrations set vp on the balance during start-
257
t
1 e -»: '
020
J!>
_^
,^
Oil!
^
,• ^
to
>
0-/
*
VO g'i ^
7^ ^
7
6f
VJJO Of
■
«i
1
ft';v
£1005
0.010
.0,015 m^
Fig. 2. 141 Aerodynamic Characteristics
of Rotor (V = 0; Mo = 0.5; o = 0.0525;
Experiment)
/gig
Fig. 2. 142 Aerodynamic Characteristics
of Rotor (Mfi = 0.075; V = 0.15; Mq =
= 0.5; a = 0.0525; Experiment).
-aoi
258
Fig. 143 Aerodynamic Characteristics of Rotor
(Mfi = 0.1; V = 0.2; Mq = 0.5; o = 0.0525;
Experiment) .
Fig.2.1M- Aerodynamic Characteristics of Rotor
(Mfi = 0.125; V = 0.25; Mo = 0.5; o = 0.0525;
Experiment) .
JO
up, overspeedlng, and nonnal operation of the rotor.
The aerodynamic forces and moments of the rotor in the axes of the wind
tunnel are determined as the difference between the readings of the balance with
the special unit operating, as well as with the unit minus rotor and hub (or with
a nonrotating hub). The obtained moments and forces include the mutual inter-
ference of the rotor with the fuselage of the unit and the effect of the rotor
hub. In some cases, corrections are introduced which take into account the ef-
fect of the rotor on the forces created hy the fuselage of the unit. In such
cases, and also when measuring with a strain-gage balance placed near the hub,
the characteristics of the rotor include only the effect of the fuselage on the
rotor and the effect of the rotor hub.
Figures 2.141 - 2.145 show the test-derived aerodynamic characteristics of
the rotor of the Mi-1 helicopter, with metal blades of rectangtilar planform.
The geometric characteristics of the blade are plotted in K.g.2.146: blade pro-
file NACA 230, number of blades z^ = 3, diameter D = 14.5 m, solidity ratio /244
CT = 0.0^25t inass characteristics of blade y = 4«5, flapping conpensator k = 0.56.
tx
^
—
-- -
—
—
\
\
k
ty'C
.20
^\^
^^
^
■^
K
^^0.16
^o.n
,^0.12
Xy=0.10
~
sjv^
s>
\^
^
*N^\
^\
y
r-'
0.01
^
y
^
V
\
■
—
-
c^=-2*^^
s\
1
-
cC=-if'
\\\
\
oC=-6°\
^.
oos
\
\
mi
r
"f
1
^
%
v\
1
1
^
-
~
—
—
^
7°
%
1
X
1
^
-0,01
OL=-
-12°^
-
'^
-
\\V
_
V^
~0,02
'
^V
I
W
-f
,
-
0.05
\
N
1
i
N
&if>
\
X
N
N
5
N
. w
-
—
N
-
If
z
f
-z
Fig. 2.145 Aerodynamic Characteristics Fig. 2. 146 Geometric Characteristics
of Rotor (Mfi = 0.15; V = 0.3% Mq = of Blade.
= 0.5; a = 0.0525; Experiment).
The escperiment was laid out so that the aerodynamic characteristics in-
cluded the effect of the rotor hub and the mutual interference between fuselage
and unit; this means that, in the aerodynamic design of a helicopter, allowance
must be made for the parasite drag of the helicopter without rotor hub.
260
In this book, we use experimental data pertaining only to the aerodynamic
design of the helicopter. Therefore, the graphs of the aerodynamic character-
istics are given for conponents of. forces in a system of wind axes ty and t^ .
The coefficients of forces t and h can tse obtained by the conversion fonnulas
[eqs.(3.15) and (3.1?)].
The rotor angles of attack, plotted in Figs. 2. 142 - 2'll+5, correspond to
zero deflection of the automatic pitch control mechanism. To reduce the voltmie
of this book, the graphs of ty = f(a, Gq , V) or ty = f(mt, Gq , V), used for de-
termining rotor pitch, are not given.
METHDDS OF CONYERTING THE AERODYNMIC CHARACTERISTICS
OF A ROTOR
The conversion formulas presented below make it possible to use e^qjerimental
data pertaining to some specific rotor for determining the aerodynamic charac-
teristics of other rotors similar. to the tested rotor with respect to dimension-
less geometric characteristics. For exanple, the experimental graphs of the
aerodynamic characteristics shown in Figs. 2. 142 - 2.145 can be used, with the
he3p of the conversion formulas, for determining the characteristics of rotors
with other soUdity ratios if the rotors have rectangular blades, a twist of
5-9°, and a profile close to the NACA 230 profile. The characteristics can be
extrapolated to other rotors, but with a lower degree of accuracy.
The use of the conversion formulas permits an appreciable reduction in the
number of graphs of aerodynamic characteristics of rotors required for helicopter
designs .
3. Conversion of Aerodynamic Characteristics to a /245
Different Rotor Solidity Ratio
let us conpare two rotors whose blades have identical distributions of
twist angles and relative chord b = -«-2 over the radius; the rotors are as-
bo. 7
sumed to have either a different nimiber of blades or a different chord' bo. 7,
i.e., a different soUdity ratio. The magnitude of the mass characteristic of
the blade y has only a minor influence on the rotor characteristics so that the
difference in y can be disregarded; however, for rigorousness we will assume
that Y of both rotors is identical.
At uniform induced velocity distribution over the disk of these rotors, the
flapping motion of the blades and all dimensionless coefficients in the body
axis system- t, h, mt, and others - are identical if the rotors have equal
values of the flight regime characteristic \j., collective pitch cp, and relative
flow normal to the plane of rotation of the rotor \ :
V sina — V
--\j.tana — v:^^a — v. (6.9)
261
It is easy to prove this from the formixLas of the Glauert-Iock theory
(Sect. 2) which indicate that the e35)ressions for all dimensionless coefficients
contain foior quantities: |Jb, X, cp, and y These quantities fully determine the
rotor kinematics; if they are equal, the velocity polygons in each rotor section
will be alike, and the true angles of attack, Cy and c^p will be equal.
Fig. 2. 147 Air Velocity Conponents Nonnal (V sin a - v)
and Parallel (V cos a) to the Rotor Plane at Different
Values of a and a.
V cos ai ~ Y cos a^ ; V sin ofi - -^^ « V sin a^ - Vg .
At known p,, X, and t, the angle of attack of the rotor is determined by the
expression
-+:
ct
tan<3.= h— =^ I /•■
(J. fi fi 4B2|Ji.-/|Jl2+ X2
(6.10)
Consequently, at equal dimensionless coefficients but different solidity
ratios, the rotor angles of attack differ; the rotor with the larger solidity,
i.e., with a larger dimensionless induced velocity, at equal t and at the same
value of A., will have a larger more positive angle of attack. Figure 2.147
illustrates the equality of the air velocity conponents normal (V sin o? - v =
= \u)R) and parallel (V cos 01 = |iU)R) to the rotor plane at different mean induced
velocities v and different rotor angles of attack Qf.
Using the subscripts "1" and "2" to denote the quantities pertaining to /246
rotors with solidity ratios of a^ and Cg > we can write the e^^jression for the
difference of the angles of attack of both rotors
tana, -tana^ = (°i-f2)<_^
4fi2^/(l2+)l2
or, approximately for y, s 0.1$ at }? 4, |j,^.
Act = C] — a2 = (oj — 02)
4B2,i2
(6.11)
The difference in the angles of attack is expressed as the product of the
262
difference in the solidities and the ratio
Consequently, at equal t
and |J, the difference in the angles' of attack is proportional to the difference
in CT.
Thus, at equal |j, and cp all characteristics of the investigated rotors in
body axes are identical if the rotors have angles
of attack differing by a quantity Aa, which is
determined from eq.(6.1l). To change over to
characteristics in a system of wind axes, we use
the formulas for converting from one system to
another. Taking into accoiint that the difference
in the angles of attack of the rotors Aor is
small, we obtain
Mrf -const
V= const
Mg- const
ty 'Const
Fig. 2. 148 Reconstiniction
of the Dependence t^ =
= f(mt) on Change of the
Rotor Solidity Ratio.
yj'
'JTj
iy^ Aa.
Let us write out the final formulas for
converting the aerodynamic characteristics of
rotors with different solidity ratios:
^1 = ^2-,
yj
^2'
/ =i J_(o oj) ^;
trif =m,.
ai = a2 + (<'i-°2)
4B2V/2
(6.12)
Equations (6.12) indi£ate that, on converting the characteristics, the
value of the coefficients V, ty , and mt is retained whereas t, and a change by a
quantity which is constant for given ty and V. This means that the reconstruc-
tion of the aerodynamic characteristics of a rotor, represented as the depend-
ence tjt = f(mt) at V = const and ty = const, reduces to a displacement of each
curve along the ordinate by a quantity At, = Act ^ — (Fig. 2. 148)'
4B^V^
Reconstruction of the characteristics need not be carried out in practice,
since we can execute the aerodynamic design of a helicopter on the basis of the
aerodynamic characteristics of a similar rotor with a different solidity ratio,
with due regard for At^ . For exanple, in designing a helicopter with a solidity
ratio CTg based on the aerodynamic characteristics of a rotor with a solidity
ratio CTi , the required torque coefficient (i^t^.f ^s ^"^ ^^^ required power N^.f^
are determined in the following sequence:
263
a) We first determine f2Un
t„ =
^2 l/2 0{u./?)2/='o2 '
F,=— •
b) For using the characteristics of a rotor vrith ai , we find
c) From the characteristics of the rotor with a-y , we determine m^ , cfi , cpi .
d) We find (m^ i)s, oi^.t^t ^.t„ s^d the required power of the helicopter:
«A/2 = «i-(''i-°2)^;
The formulas derived above were obtained for rotors with uniform induced
velocity distribution over the disk. For rotors with an infinite number of
blades and nonuniform induced velocity distribution, no argiunents or conversion
formulas would change if at each point of the disk the induced velocities of the
rotors with different soUdity ratios differed by an identical quantity equal to
VAof. In reality, the change in solidity ratio influences also the variable com^
ponent of the induced velocity, i.e., the induced velocity diagrams do not differ
by the same quantity. However, since we are converting the average per-revolu-
tion characteristics ty, t^, and m^, which are mainly determined by the average
portion of the induced velocity, the conversion formulas (6.12) can be used with
stiff icient reliability.
The proposed method of converting aerodynamic characteristics of rotors is
similar to an analogous method for airplane propellers. It is also based on de-
temiining regimes in which the kinematic characteristics of the propellers with
different solidity ratios are identical. The difference is that, for propellers
with different a, different flying speeds (different \~) are chosen, whereas
for rotors for which the kinematics is determined not only by the normal veloci-
ty but also by the velocity conponent in the plane of rotation, the flying speed
is retained but different angles of attack are chosen.
264
4« Conversion o f Aerodynamic Characteristics on Variation
in Minim um Profile Drag Coefficient of the Blade
Sections c.
If the blades differ in magnitude of the minimum profile drag coefficient
[different quality of manufacture of the profile (see Sect .4.3), different /2i?i-8
profiles differing mainly in c^pg], the following formulas can be used for con-
verting the aerodynamic characteristics of the rotors.
The increments in the coefficients of torque ajid longitudinal force, accord-
ing to the Glauert-Lock theory, vdll be equal
to
W// = const
V=const . Ac,
M„= const
ty= const
Affi( =
f-xp
(1 + 1^^);
AA=-
Ac
xp
^,
(6.13)
(6.14)
where Ac
xp
= (Cxp„)2 - (CxpJl •
Po'
Increased
The remaining coefficients in the body
axis system remain unchanged. The coefficient
tx increases by an amount equal to about Ah.
Thus, the conversion formulas will be
V2-Vt;
^2"
Fig. 2. 149 Reconstruction of
the Dependence t^ = f(mt) on
Variations in the Minimum
Profile Drag Coefficient of
the Blade Sections.
yr
^-2 = ^x,+4-Ar,,V-;
?2 = ?l.
(6.15)
The variation in the rotor characteristics •upon conversion is illustrated
in Fig.2.1!t.9.
ty t.
Conversion can also be performed, provided there is constancy of the quanti-
^2=^yi'
ix =tr ■
Under this condition, the increment in torque will be equal to [see
eq.(3.7l)]
A m^ = Am;,r =^ Ac^p ( 1 + 31/2)
265
and, accordingly,
'nt,=mt^-\-~Ac,^(l+3V'). (6-16)
By analogy with eq.(3«72), in place of eq.(6.l6) we obtain
Using eqs.(6.l6) or (6.I7), we change from point 1 in Fig. 2. 149 to point 3.
The nonrigorousness of the conversion formula lies in the fact that, while
it does take into account the effect of an increment in Cj^p , it disregards the
fact that, at points 1 and 3, the angles of attack of the rotors are not mutual-
ly equal (a^. = 0/2 ; cxs < ffg ) . Consequently, in these regimes there is a differ-
ent distribution of the true angles of attack of the sections and therefore dif-
ferent induced and profile powers at identical profile polars. However, the /2L9
conversion methods are approximate so that the indicated inaccuracy is of no
practical value.
If the polars of the profiles differ not only in the quantity c^p but also
in their slope, the proposed method will not be valid. Therefore, it is unsuit-
able for converting aerodynamic characteristics to other Re and Mq numbers. In
these cases, the quantity Am^ should be determined with consideration of the real
values of a^ and c^p at each point of the rotor disk.
5. Conversion of Aerodynamic Characteristics on Variation
in the Peripheral Speed of the Rotor (Mq Numbers)
Fig-ures 2.80 - 2.88 in Section 3 give graphs for the increment in torque
coefficient at Mach numbers greater than 0.4* The graphs show that at moderate
values of the thrxist coefficients the conpressibility of air has a noticeable
effect on the quantity mt at Mq greater than 0.55 - 0.6. Therefore, vpon a
change in Mq beyond these limits (and at near-separation values of ty at lower
Mq) corrections must be introduced into the aerodynamic characteristics of the
rotor. These corrections are determined from the graphs in ELgs.2.80 - 2-88 as
the difference of the values of Amoo at the Mach numbers in question.
For example, if the experiment was carried out at Mq equal to Mq^ , and the
experimental data are used at Mq equal to Mq , then the value of m^ = mt(V,
ty, tjt, Mq ) found from the e^qjerimental graphs, must be stpplemented by
8m,„ = Am^^(Mo^)-Am^^(Moj), (6.18)
where Amoo a^re determined at corresponding Mq numbers and at the same values of
Thus,
266
mtiV, ty, tjc, ^02)='"^;,^ (^. (y, tx, Mo,) + Sm,:,,. (6 .19)
At V > 0.3 when Mq changes, we must introduce a correction to the angle of
attack of the rotor and, accordingly, to the blade pitch. These corrections are
introduced analogously:
a(^, ty, tx, ^^)^^<iyp (^. ^y. ^x, Mo,) + K^; (6.20)
8a^<,=Aac^(^, ^y. ^jt, Mo2)-Aa iY Jy,t^,Ni^^. (6.2l)
The blade pitch is determined in relation to V, ty, and 01 from the graphs
of ty = f(ff, 9o , V) on the assung^tion that this dependence does not change with
respect to Mq.
6. Conversion of Angle of Attack and Rotor Pitch on
Variation in Inclination of the Automatic Pitch
Control. Flapping Compensator, and Mass
Characteristic of the Blade
As shown in Section 2, a change in the slope of the automatic pitch control
and the flapping conpensator will not cause a change in the coefficients ty, t^,
and m^ provided that the equivalent rotor angle of attack cv, = a - cpx remains as
before. Consequently, conversion of the rotor characteristics reduces to find-
ing the new rotor angle of attack by means of the expression
a2 = ac+(?i)2 = ai- (?i)i + (9i)2. (6.22)
where 91 is determined by the formula /250
Cpi = — DjX -|- D{(\ — k
l+ft2
The coefficients ai and bj. are found from known values of ax and bx by
means of eqs. (2-273), (2.274), (2.249), and (2.250). The rotor pitch is con-
verted by the formula
6o2=?a + Mo=S~"')(*'~*2)- (6.23)
On variation in the quantity —^ — , it can be considered that the coeffi-
cients ao and bj^ entering eqs. (6. 22) and (6.23) vary in direct proportion to
v
the ratio of the new and old values of -^ l -, x.e,,
"•inn
267
In conparing eD^jerimerital data with each other or with calculated data, one
miist also account for the effect of the rotor hub, as is done in the exaiiples
given below.
?• Examples of Using the Conversion Formulas
Comparison of calculated aerodynamic characteristics with espjerimental .
The calciilated aerodsmamic characteristics of a rotor with rectangular blades
a = 0.0525 were obtained by the method presented in Section 4* The only differ-
ence between calcxilation and esgjeriment, which should be taken into account when
making a conparison, is the effect of the rotor hub on the esqaerimental charac-
teristics. Taking "c^^ = 0.0015, a reduction of the calculation to the e^qjeri-
mental conditions requires the addition of the following increment to the calcu-
lated propiiLsive force of the rotor:
or, in dimensionless form.
^^-^=^-^^^ ^
A^,, =c; z!. (6.24)
Thus, the e^^jerimental curves can be conpared with the quantity
_ _|_ 2:0015 1/2^/ +0.02861/2.
*ca/c ' 0.0525 'ca/c '
In Fig.2.150, e^qserimental curves and converted calculated curves {t^) are
plotted for V = 0.3. The diagram indicates that, in powered flight regimes
(tx = -0.01 0.02), the difference in m^ is negligible whereas, in autorota^-
tion regime, the values of t^ differ by 5 - 15^ • The convergence of the experi-
mental and calculated curves is better at lower V.
Comparison of experimental aerodyn^ic cha racteristics. Using the experi-
mental aerodynamic characteristics of a rotor with trapezoidal blades of /251
NACA 230 profile, plywood planking, a = O.OS65, and Mq = 0.4, the conversion
formulas will yield the characteristics of a rotor with rectangular metal blades
with NACA 230 profile, a = 0.0525, and Mq = 0.5, which can then be conpared with
the experimentally obtained characteristics shown in Figs. 2. 141 - 2.145 •
The difference in the solidity ratio is taken into account by eq.(6.l2):
<; (.=0.0525) = (' ('=0.0865) + (0.0525 - 0.0865) -^ =
^tx (0=0.0865) — 0.009 =|- .
268
0.01
-0.01
-0.02
-0.02
>
N
h
-
Experiment
Conversion
^
N
y
^
V
^
^
-
\
^\
N
\
\
\
v
.
w
\
\
\
■ -
\
w
\
^
0.010
mt
\
s3
V
ty'
?-^?d^
^
u.ia
1^=0.10
^
^
\
\
"=^
-^\^^
1
w
\
., .
-
\
V
V
s
\N
k
\\\
\
\
v
V
V
K.g.2.150 Conparison of Experimental Iilg.2.151 Ccaiqsarison of Esqjerimental
and Calculated Aerodynamic Character- and Converted Aerodynamic Characteris-
istics of Rotor (Mfi = 0.15; V = 0.3; tics of Rotor (M^ = 0.15; V = 0.3;
Mr
0.5; cj = 0.0525).
Mr
0.5; a = 0.0525).
The difference in the profile power of the rotors, with consideration of
the difference in tlade planform, is found from the following expressions:
m
p>-
,,=0.0525)=— ^(1+5K2)P,;
nipr (0=0.0865)
"xp.
Ac,
LJnpr =
Ac
xp
■«*'(i+5p2)p+^^::^
'■xp.
(l+5\/2)P;
(6.25)
(1+5K2)P ^ (1-1- 51/2) (p_p^)
Admitting for the trapezoidal blade P = 0.92, c^p = 0.009 and in con-
formity with the recommendations in Section 4«3, ACxp ~ 0.0025, we find
Am^;.= -|'5^0.92+5i02? (0.92- 1)1(1 +5V^2)= -0.0004(1 +5K2).
So as to make the conversion of the characteristics only with respect to /252
one of the coefficients, namely with respect to t^ at mt = const, we convert
269
Anipi. to Atjt .
M,^^^^. (6.26)
The difference in the Mq mjmbers is taken into accoxont by eqs.(6.19), while
Anioo is determined from the graphs in Figs. 2. 80 - 2.88.
At a difference in solidity ratios, the hub drag resxilts in a different in-
crement of the coefficient t^ :
(^^J.=^'^^=
Cj-
To convert the characteristics, we must subtract (At, )x from the coeffi-
hub
cient of the propulsive force of the e^^jeriment under conversion (oi ) and add
(^■tx....)2:
*hut ■
'"■.>=''>-~^-{^i,r+{¥lr- ^'-^^
Since, in analyzing the experiment with trapezoidal blades, the drag of the
nonrotating hub was excluded, we can take c^ = 0.00075 to account for hub /253
rotation. Therefore, the conversion for the effect of the hub is performed by
means of the expression
. . / 0.00075 0.0015\t7,
^ (.=0.0525) = /. (.=0.0865) -^-^^^^^^ ^^^^)V =
= /:r (.=0.0865) +0.02\72.
Thus, the final expression for converting the coefficient t^ has the form
ix (0=0.0525) ^tx (0=0.0865) — 0.009 ^|
_ 0.0004 -^^tj^ I 0.02 K2 4- ^ .
For conparison purposes. Pig. 2. 151 gives the e^qjerimentally obtained and
converted (t^) characteristics. For the most part, the agreement of the curves
is satisfactory.
Section ?. Performance and Propulsive Efficiency Coefficient
of a Rotor
The helicopter rotor produces lift and simultaneously acts as the prime
270
mover of the helicopter. Therefore, it is natioral to characterize its lifting
and propiilsive properties in the same manner as a wing is characterized by the
performance K„ and a tractor propeller by the efficiency Tlt.p • These concepts
permit definition of the degree of siiitability of a rotor as a means for produc-
ing lift and propulsive force, as well as a rapid performance, in general form,
of approximate calculations of the required power of a single-rotor helicopter,
a helicopter with a wing and tractor propellers, or a multirotor helicopter, and
proper selection of the regime of maximum performance (maxim\m range). Knowing
the performance and efficiency, one can estimate directly the expediency of in-
stalling a wing and tractor propellers on a helicopter, determine what part of
the total drag of a helicopter is made vp by parasite drag and how much the re-
quired power can be reduced when the parasite drag is reduced, and find the
rational distribution of power between tractor propellers and rotor.
A determination of rotor performance in an autorotation regime is carried
out in the same manner as for a wing. The concept of rotor performance has been
widely used in aerodynamic designs of autogiros. The rotor performance, together
with a coefficient which we will call the propulsive efficiency coefficient, can
be used also for calculating a helicopter, as we will demonstrate below.
Unlike in an airplane, where the wing and propeller are different units and
K„ and \,j, can be examined independently of each other, in a helicopter the
rotor performance K ajid the efficiency T] are interrelated and the efficiency of
a iTotor in any regime is determined by the value of the product KT].
Let us first discuss the concepts of performance and efficiency, described
in individual works on helicopter aerodynamics (K.Khokhenemzer and other
authors) .
1. Performanc e and Efficiency of Rotor Proposed /254
by K.Khokhenemzer
Rotor performance can be determined on the assimption that the actual pro-
piilsive force of the rotor (-X) is the difference between the ratio -l and
some arbitrary drag of the rotor Xaj.Tj\^-X = -^ X^rbj* from which we determine
the arbitrary drag of the rotor:
Correspondingly, the rotor performance is
y r
+x
t/'^^^
Xars ~'75N . ^ ■ (7.1)
V
The ratio —^ — would be equal to the propulsive force if the entire power
were converted without losses into propulsive force. Since the actual propulsive
271
force is equal to —^ ^art* ^'^ ^s oTdvIous that all losses belong to X^^ .
Thus, the rotor is represented as a certain mechanism creating forces Y and X^ri, ;
the power si^jplied to it creates, without losses, a propulsive force equal to
■ 1^ so that the total (actual) propulsive force is equal to '^ ■ ■ - X^^^,, .
The arbitrary drag of the rotor is conparable with the drag of a wing plus
the power losses of the tractor prcipeller, i.e.,
while the total propulsive force of the system "wing + tractor propeller" is
equal to ■ '^^y^'" -X„^^.
As a second version it is proposed to consider that the lift of a rotor is
produced without loss (without drag) and that all losses are accounted for by
the generation of a propiolsive force. The rotor is represented as some mechanism
producing lift Y, while the power sipplied to it is converted into propulsive
force
Hence,
,,==^=1=^. (7.2)
75iV m^
It is obvious that the efficiency T] cannot be conpared with Tlt.p but with /255
the difference Tl^.p =— ^ -» since the thrust of the propeller minus the
wing drag is equal to "
'tE.
The description of both versions of representing the characteristics of a
rotor shows that they are both artificial and conparable only with combined
characteristics of the wing and tractor propeller. This is the adverse side of
the proposed concepts. Their favorable side is that the characteristics are de-
scribed only by one quantity: either by performance or by efficiency.
The concepts of rotor performance (or efficiency) examined above are con-
venient for calculation, since they relate flying speed V and helicopter weight
G (or propulsive force) with the required power. Actually, having set in
eq.(7.l) G = Y and Qpar = -X, we obtain
272
^^•/=^(f+^.'>
(7.3)
However, the sense of applyiiig these concepts is predicated rpon the con?-
venience of use in calculation and in determining optimal parameters. Otjt con-
cepts presented below also simpHiy the calculations and, furthermore, while re-
taining the sense and value of analogous con-
cepts for airplanes, facilitate an investigation
V- const Mo^const °^ conposite rotary-wing aircraft.
e= const
Y= const
2. DeteiTiiination of Performance and Propulsive
Efficiency of a Rotor
Let the rotor operate in the regime "a"
(Fig. 2. 152) with a propulsive force Xa, requir-
ing a power N^. To increase the propulsive
force to X^ while retaining the lift Y, the
rotor must be tilted forward and the power must
be raised to N^. The efficiency of the rotor as
a propulsion unit on changing from the regime
"a" to the regime "b" is defined as the ratio of
the power increment of the rotor as a prime
mover -AXV to the increment of power stpplied to
the rotor:
Fig. 2. 152 For Determining
the Concepts of Performance
and Efficiency of a Rotor.
Tl = -
— IlXV
-At^V (t,^~t,^)V
75\N
Am».
nif —ntf.
(7.4)
If, to increase the propulsive force of the rotor we were to install a
tractor propeller and supply it with a power equal in magnitude to the differ-
ence N^ - Na = AN, then it would create a thrust of
(7.5)
A conparison of this e^q^ression with eq.(7.4) shows that the incitement in
propulsive force of the rotor -AX is characterized by its propulsive efficiency
T] just as the thrust of a tractor propeller is characterized by its effi- /256
ciency Tlt.p •
In a craft with a rotor installed to produce lift, the power can be sxpplied
either to the rotor (helicopter: Nt.p = 0, N^ot = %* see Fig. 2. 152), or to the
tractor propeller (autogiro: N^o^ =0, N^.p = N,,), or distributed between the
first (Ni.ot = Na) and second (N^.p = N^ - Na). A comparison of T\ with Tlt.p
shows which of these versions is better, i.e., whether it is expedient to install
a tractor propeller for increasing the propulsive force of the craft or whether
it is more advantageoiis to transmit the entire power to the rotor: if 1) > Tlt.p,
then |Xi,| > |Xa| + Pt.p » More precisely, we must ccaipare 11 with Tlt.p
§
VjjE.
273
(the ratio *'P takes into account the difference in losses of power trans-
mitted to the rotor and to the tractor propeller) •
Thus, in order to obtain for a rotor, which is a lift-producing coriponent
as well as a propulsion unit, a coeffi-
cient analogous to the efficiency of a
y tractor propeller, it is necessary to
investigate the increment in propulsive
^ _v. force (or drag) of the rotor when power
^ — is supplied to it. Therefore, we de-
^ ' 1 fined the propulsive efficiency of a
-X
N
■ Direction of flight ^ rx-x)y Totor 33 the ratio of the incromonts of
• " /(=-i- Tj= ^ <^ '^ useful and expended work, although such
'"''^ ^^'^ a ratio is not actually the efficiency
but only performs its role for craft
Fig. 2. 153 Model Representing a with a rotor.
Rotor as Two Elements - lifting
and Propelling. To determine the propulsive effi-
ciency with respect to eq.(7«4) we must
select some regime as the initial (we
select here the point "a"). The drag of the rotor in this regime X^ determines
its performance.
It is ej^edient to take, as initial regime, an autorotation regime (point
"c" in Fig. 2. 152). in this regime, no power is supplied to the rotor which, in
producing lift, also creates drag like a wing.
Thus, the work done Ipy a rotor can be interpreted as follows: The lift is
generated by the rotor in an autorotation regime without the expenditure of
engine power, just as for a wing; in regimes with a supply of power the rotor
creates a propulsive force which partially conpensates (at N < Nj,) or overcomr-
pensates (at W > Nj^) the rotor drag in an autorotation regime. The propulsive
efficiency characterizes the power losses of a rotor when changing to an engine
(prcjpulsion tonit) regime. The rotor is replaced by the model shown in Fig. 2. 153,
for which, in conformity with the foregoing, the expressions for T] and K have
the form
■^- ,n, ' (7-6)
••c
In eqs.(7.6), (7»7) and below the subscript "c" means that the indicated
quantity refers to an autorotation regime.
In level flight, the propulsive force of the rotor is equal to the sum of
the rotor drag t^ and the parasite drag of the helicopter t^ - t^ = t^ + /257
+ "Cjj ~— — . Consequently, the propulsive efficiency and performance of a heli-
copter in horizontal flight are equal to
274
K,
or
Ku
GV
h,-^ 75Ar,_^r,
(7.8)
(7.9)
(7.10)
We note that the quantity (t^ - t^ ) represents the coefficient of the
arbitrary propulsive force of the rotor in horizontal flight and is equal to the
coefficient of the drag counteracted by the tractor propeller of an autogiro or
helicopter for which, in horizontal flight, the rotor operates in an autorota-
tion regime.
V = const, M„-const
e = const
ty= const
U=const Mg=const
6= const
tyconst
%^ i^="«^
K
V*'.
V
Fig. 2. 154 Determination of Rotor
Efficiency at linear Dependence
of tx on m^ ,
Fig. 2.155 For Estimating the Expedi-
ency of Installing a Tractor Pro-
peller on a Helicopter.
In eqs.(7.7), (7.9), and (y.lO), the quantities K and K^ are the same as
those used in autogiro calculations.
Use of the concepts of performance and propulsive efficiency for calcular-
tion is highly convenient in the case of linear dependence of t^ on m^ . Here,
the quantity 71 does not depend on m^ or t^, since the ratio — jr- is equal to the
angle of slope of the straight lines t^ = f(mt) (Fig. 2. 154).
In place of the aggregate of the graphs (see Figs. 2. 105 - 2.109) constructed
for several V, the aerodynamic characteristics can be represented as two graphs:
K and 7] as a function of ty and V (see Fig. 2. 158), by means of which t^ is
275
determined from eqs.(7.7). and (7.6) at known values of ty, V, mt, or else m^ is
determined at known ty, V, t^^.
In the case of a nonlinear dependence of t^^ on mt, when T) depends on mt,
the use of K and T) in calculations offers no substantial advantages. Here, it
is of interest to determine the propulsive efficiency with respect to the /256
angle of inclination of the tangent to the curve t^ = f(mt) at the point in
question
^1 =
dm J,
(7.11)
A conparison of Tli with Tlt.p i'^ permits determining whether the propul-
sive force of a craft can be increased by installing a tractor propeller. Let
V'ConstiMg'Const
i^=const
■f]^ = 7]^^^ 5^.T> at the point "a"
It is obvious that, if
(Fig. 2.155).
mt < mt^, we have Tli > Tit.,
%LL- and use
^=-'
Fig. 2. 156 For Determining the
Performance and Efficiency, at
Nonlinear Dependence of t^ on mt >
of a tractor propeller is not advantage-
ous. If the rotor operates in a regime
with mt greater than mt , the installa-
tion of a tractor propeller may increase
the propulsive force, the maximum gain
being obtained when a power corresponding
to mt is transmitted to the rotor and
to the tractor propeller (mt - mt ) .
In the case of a nonlinear dependence
of tj on mt , it is preferable, for ap-
proximate calculations with the concept
of performance and efficiency, to replace
the nonlinear dependence of t^ on mt by a linear dependence. Such an approxima-
tion is made in the segment of the straight lines from a = -20° and sometimes
from a = -15° at V = 0.15 (point F in Fig. 2.156) to the minimum value of mt
(point H in Fig. 2.156) at which the greatest deviation of mt from exact values
does not exceed 3% of mt at t^ =-0.1 V^ (approximately a horizontal flight
regime of helicopters) .
The value of T] and K, determined from the approximating segment HF, is
calculated by the formulas
V
K=^=-
mt^
(7.12)
(7.13)
276
The efficiencv detennined by the angle of Inclination of the linearized de-
pendence tjt = f(mt; must be regarded as the propulsive efficiency, on the aver-
age, for the cturve.
3« Performance and^Efficiency of a Rotor, Obtained
from Experimental Data
The graphs for the aerodynamic characteristics of a rotor in the form of
the dependence t^ = f(mt) obtained from experiments in a full-scale vdnd tunnel
are given in Section 6 in Figs. 2. 142 - 2.145 • They pertain to a three-blade
metal rotor with rectangular tvdsted blades, a = 0.0525, and include the drag of
the rotor hub. From these graphs and using eqs.(7.6), C7.7), we determine the
dependence of K and 11 on ty shown in Figs. 2. 157 and 2.158.
n
w
=
-
■
—
■^
-^
.
i
t
iZ^^J
no
. ^=0.75
n?.?
^
0.84
t
>F-
■R?0
0.20 ty
0.10
ais
OJZO ty
Fig. 2. 157 Rotor Performance according Fig. 2. 158 Rotor Efficiency according
to Experimental Data (Mq = 0.5; to Experimental Data (Mq = 0.5;
a = 0.0525). o = 0.0525).
To define the character of the slope of the curves of K, let us examine /259
the approximate expressions for K. According to the energy method of calcula-
tion (Sect .3), we have
1
V
1
f<ind
\pj.
'^ind
^Pr.
K
V 4V2
(7.14)
(7.15)
(7.16)
pr
277
The increase in performance with an increase in V can be attributed to a
decrease in induced and profile drags with an increase in V. At average values
of ty, the performance depends little on ty since the induced part of the re-
ciprocal performance increases with increasing ty whereas the profile part de-
creases (tp to incipient flow separation). At small ty, the performance de-
creases owing to an increase in
Kpr
The maximum magnitude of rotor performance depends on V, Mq, ct, quality of
blade maniifacture, and geometric blade characteristics . Optimum performance was
not obtained in the experiments, owing to the low value of Vj,ax • The largest
of the obtained values is K„ax = 9*7 at V = 0.3, ty = 0.1?-
Rotor performance is lower in magnitude than wing performance. This is ex-
plained by the fact that a rotor has greater profile losses than an airplane
wing since, at equal flying speed, the flow across the blades has a much greater
velocity U. In the case of undeflected mechanisms, the profile drag of a wing
is by a factor of 2 - 2.7 less than that of a rotor. Upon deflection of the
mechanisms, the profile drag of a wing increases appreciably and approaches the
profile drag of a rotor.
A rotor and wing are closely adjacent in value of induced drag (at t„ = D
and at uniform induced velocity distribution, the induced drag is the same) .
A decrease in performance at small V is inevitable both for a rotor and /260
for a wing, owing to an increase in induced drag. However, the wing cannot have
as low a £^erformance as a rotor, since the wing cannot have as high a Cy as a
rotor at V < 0.15 [see eq.(4.37), Chapt.IIl].
The propulsive efficiency of a rotor varies within the limits of 0.99 to
0.9* The curves of T] intersect one another, and in some cases there is an ap-
preciable scattering of the test points. The fact is that it is difficult to
determine accijrately the quantity T], since the scattering of the test points on
graphs of t^ = f(mt) creates some indeteradnacy in the angle of slope of the
straight lines, which has a noticeable (within 3 - 5%) effect on the quantity T\.
The inaccuracy in determining T], and also K, shows that when representing
the characteristics of a rotor in the form of lifting and propelling elements it
is inpossible to estimate them separately with high accuracy. However, this
does not mean that calculations performed with the use of K and T] have a low ac-
curacy, since when determining m^ by the formula
(^-'-)
V
„ _, .__ (7.17)
the errors in determining T] and K are conpensated .
Figure 2.158 shows that, even if the low accuracy of determining the rotor
efficiency is taken into consideration, this efficiency is greater than that of
a tractor propeller. Since 7] is defined as the ratio of the increments of use-
ful to e353ended work, it need not be less than 1.0. We will e^gjlain this. let
278
us substitute into eq.(7»6) the expression for rotor power taken from the
energy method of calculation
75N=15Npr-^15Ni„j~XV. (7.18)
Then eq.(7«6) takes the form
T] =
(Xc -X)V + 7h {Nm - W,We) + 75 (V-^^) " ( 7 -19 )
It is clear from eq.(7»19) that, if the induced and profile powers of the
rotor which depend mainly on lift were not to change on a variation in the pro-
pulsive force, then the propulsive efficiency wotild be equal to 1.0. Actually,
the differences of N^^ and Np^ are small, since we are examining the change of
propulsive force at constant lift and flying speed, i.e., at approximately
identical average values of induced velocity and true angles of attack of the
blade sections. We can show that, for an identical propulsive force, these dif-
ferences are respectively smaller than the induced and profile powers of a
tractor propeller, as a result of which T] > llt.p •
Thus, in examining the lifting and propulsive properties of a rotor, we de-
termined that the bulk of power losses (Nj^^ and Np^ in an autorotation regime) is
accounted for by energy losses related with the production of lift, which deter-
mines the low performance of a rotor.
The propulsive efficiency of a rotor differs from 1.0, owing to the small
difference in induced and profile losses in regimes with power supply to the
rotor and in autorotation regimes; it is greater than the efficiency of a
tractor propeller.
It should be borne in mind that the values of K and Tj, whose dependence /261
on ty is shown in Figs. 2. 157 and 2.15S, are valid for regimes within limits in
which the e^qDeriments are carried out. This means that mt, calculated by
eq.(7.17), can be correctly determined, if it is not greater than the maximum
values of mt up to which the experimental curves were plotted (mt
= 0.01 - 0.013). "*''
4. Performance and Efficiency of a Rotor. Obtained
from Calculated Graphs
The performance and efficiency of a rotor with rectangular twisted blades
(variant II in Table 2.10), a = 0.091, were determined from graphs of the aero-
dynamic characteristics obtained by calculation. In the case of nonlinear de-
pendence tj = f(mt), the quantities K and T) were found from eqs.(7.l2) and
(7.13). The graphs of K and 1] are shown in Figs. 2. 159 and 2.160.
Rotor performance begins to decrease at Mg > 0.6, especially at large Vj at
Mo = 0.7 and V = 0.3, K diminishes by 1.5, and_at V = 0.4 by 3.5. At Mq = 0.7,
the performance at V = 0.3 is greater than at V = 0.4, and the maximum performr-
ance is equal to about 7.5.
279
The efficiency of a rotor, for Mq =0.6 - 0.7 at average and small values
of ty, has a higher value (more than 0.95). At near-separation values of ty,
the efficiency begins to drop markedly, but does not decrease when ty = ty
less than 0.75 - 0.85.
K
■^
^
_
M
-.nil
"
^^
v=u.w
W
,-
^
^
1
%=ft6
9
,
^
^
_
J
M,=0.7
_,
^
~
—
■-
8
/
^
'
^
Y
V'0.30
/
^
^
^
7
/
'/
^
^
1 — '
i—
I—'
"
""
--
-^
/
C^
^
-
""
V'OAO
6
->
''
^
,^
^
v
^
-J
J
M 'QAandO.G
5
^■
r
,_j
^
[
, ,.,
~~
1 1
r— n
U-
^
"^
--
-
"
'-
-■
—
~,
.^
^
' 1
f
i>
'
-
/=fl20|
t
^
^
«
^
^,
,M=
3
-
—
-
—
-
—
-
—
—
—
-
-
-
-
"=^^
;;— 4-
2
IF n.„rr\ 1
_
_
¥ •
-U.IO' J
0.08 0.10 m 0.1 1 0.16 0.18 0,20 0.22 0.21 ty
Fig. 2. 159 Rotor Performance (Calculation, a = 0.091).
The values of K and Tl obtained as a resiolt of linearization of the curves
of t^ = f(mt) hold true within certain limits. The tpper limit of applicability
mt / 7CK
of the graphs of K and T] are the values of the ratio — "ii'- = V~YV
given in Table 2.13- "^
7
1.0
a95
0.90
ass
0.B0
fl7J
~
z
11
—
"^
^
■r-
—
^
--
^-.
—
.^
I'
■_
r=
—
-~
~^
L.
L
-t-
—
■ —
--
>=o
—
-•
■
—
_.
w^o
.7
zo'
-
p.
-=:
~
' —
—
L.
Ij;;
■-
-^
Vmni,
■^
^
Sr
ti
■^
~"
*^
E^
—
—
■^
k-i
*^''-
<5
■v
^
K
i _
\
\
\
\
^
^,
^'
^
M
r--
«5
».^
<^
*
h>
>.■
\
"^
^
-<
s
\
Xx
■V
s
s
^
>
\
\
\
\
s
\
^
\
"
\
s
^
\
>
\
c
\V
N
s
\
\
N
\
%
s
N
i
\
\
-
-
\
V
_
\
^
\
\
\
\
\
1
^^-^
\
^\
\
t
-
\
\
\
\
-
.^\
^
-
\
.s>
VXr
I
I
\
^\
^
M„=0.6
*
\'-
M =07
\
• X
_
^J
OM 0.10 012 p.l't 0,16 0.18 aZO 0.22 O^/fty
Fig. 2. 160 Rotor Efficiency (Calculation, o = 0.091).
280
If the ratio
75N
YV
obtained from calculation is less that that given in
Table 2.13, the calciolation will not differ by more than 3% from the calctalation
made from graphs of aerodjmamic characteristics. Furthermore we note that the
75N
values of
YV
given in Table 2-13 correspond to flight regimes with a = -20° /262
(sometimes -15° at V = 0.15).
The lower limit of applicability- of the graphs in Figs. 2. 159 and 2.160 is
the autorotation regime or a powered glide. More accurately, the calculations
for an autorotation regime are performed from the graphs of K^ shown in
Fig. 2. Ill, since the values of K found from the linearized curves may differ
somewhat from K^ .
TABIE 2.13
/ 75/V \
V YV )
0.15
0.6
0.2
0.48
0.3
0.37
0.4
0.3
A Gonparison of the performance and efficiency of rotors having blades with
different geometric characteristics shows that, at Mq = 0.4 - 0.5, the blade
profile influences the value of K to within several percents whereas T] depends
little on the blade profile. This means that rotors with different profiles re-
quire a power differing by AN, where AN is independent of the type of operating
regime of the rotor, namely at either large or small propulsive force (in gliding
or climbing).
For trapezoidal blades (variant I of the blades in Table 2.10)_j_ K is greater
by 0.5 (at V = 0.2) - 1.5 (at V = 0.4), and T] is lower by 0.01 (at V = 0.2) -
- 0.03 (at V = 0.4) than for rectangular blades. This means that the greatest
decrease in required power for a rotor with trapezoidal blades occurs at small
propulsive forces. At large propulsive forces, the rotor with rectangular /263
blades having a larger 1\ may prove to be better.
Conparative graphs of K and 11 at Mq = 0.7 are shown in Figs. 2.161 - 2.164.
The diagrams show that, for a rotor without a high-speed profile at_the blade
tip (variants III, IV), K is smaller by 0.7 (at V = 0.2) - 1.7 (at V = 0.4) than
for a rotor with a high-speed profile. For trapezoidal blades, K is higher by
0.5 - 1.0 and T| lower by 0.02 - O.OS, respectively, than for a rotor with rec-
tangiilar blades.
For a rotor with blades of increased geometric twist (variant VI) and with
blades e^qjanding toward the tip (variant VIl) at V = 0.4, the performance is
0.5 - 0.7 lower and the efficiency 0.05 - 0.15 higher. The very high value of 11
281
J\
for a rotor wLth increased twist is a consequence of the substantial decrease
in profile losses tpon an increase in propulsive force of the rotor (see
Pigs. 2. 75 and 2.76).
-
_
1
_
7 ^
-
^
—
■^
ill
1
pj
0.W 0.12 0.11* 0.16
0.18
0.10 0.12 O.l't 0.16 0.18 ty
Fig. 2. 161 Performance of Rotors with Fig. 2. 162 Efficiency of Rotors with
Blades of Different Shape (V = 0.2; Blades of Different Shape (V = 0.2;
Mo = 0.7). Mo = 0.7).
K
' —
—
^-
--
1
1
/?
6
^
^
--
m
'
^
--
1
-■
•--
--
5
-^
---'
^
"
---
-
—
"
111
■—■
'
'
■s
1
~~
^
X
^
V
^
\,
-
:
^
^
s
1.0
-—
■X
"-
_.
r:
-
-
-
030
-
—
'
\
nsn
^
VII
aw
ai2
W* ty
aw
0.12
Fig. 2. 163 Performance of Rotors with Fig. 2. 164 Efficiency of Rotors with
Blades of Different Shape (V = 0.4; Blades of Different Shape (V = 0.4;
Mo = 0.7). Mo = 0.7).
5. Conversion p_f Pe rfggngi-Iice and Efficiency on
Variations in^Rot or Parameters
In conformity with the formulas deri-ved in Section 6, the performance of a
rotor on variations in the soUdity ratio and profile power coefficient is con-
verted by the expression
1 (a — oj) ty Am^yti
(7.20)
/< /Ci
4B2K2 ' tyV
282
A change in profile power coefficient should take into account a change /264
in Cxpq of the profile (or Ac^p , owing to the difference in the quality of blade
manTrCacture) and a change in mp^ from the wave drag:
Mn.
Ac_
•(l+5l/2)P + 5/n„
(7.21)
The propulsive efficiency is independent of the difference in a and Ac^p .
The quantity bva.^^ depends on t^ so that also T] depends on t^ •• However, for the
sake of singjUcity w6 need not convert T], and we substitute 5moo into eq.(7.2l)
at an average value of t^ •
6. General Conim ents on Rotor Efficiency and Performance
Figures 2.165 - 2.16? show the generalized graphs of K and T], which can be
used for estimate calculations. Figure 2.165 gives the graph of K for a =
= 0.091, which is valid for average and_large ty . At small (ty ~ O.l), K is
smaller by 0.2 (at V = O.I5) - 1.5 (at V = 0.4). Figliore 2.166 contains the
graph of 11 used for all ty at Mq ^ 0.55- For Mq > 0.55, the efficiency must be
corrected by a quantity AT], which is plotted in Fig. 2. 16? as a function of ty.
M,
o>
V.
0.1
Rectangular blade
Trape zoidal blade
LI I \'\ 1.1 rm I
0.2 0.3
0.8
~ Trape zoidal blade
li Jill rn jjn
0.1
0.2
a3
Fig. 2. 165 Generalized Graph of Rotor Fig. 2. 166 Generalized Graph of Rotor
Performance (a = O.O9I). Efficiency (Mq s 0.5).
Thus, as shown in Figs. 2.165 - 2.167, the rotor performance is lower than
the wing performance, and the propulsive efficiency of the rotor is higher than
that of a tractor propeller. This is e^iplained by the fact that the bulk of the
power losses pertain to losses related with the production of Hft, whereas the
propulsive efficiency differs from unity owing to the small difference in in-
duced and profile losses in regimes with power sipply to the rotor and in auto-
rotation regimes.
Thus, it is obvious that the installation of a wing with a performance
higher than that of a rotor will increase the performance of the lifting system
of a helicopter. The installation of a tractor propeller of an efficiency lower
than the propulsive efficiency of a rotor will lead to some increase in required
283
/>'?|0.T0 0.1Z O.l'f 0.16_ 0J8 0.20 0.22 ty
power. Therefore, a tractor propeller on
a helicopter can be usefiol when the relief
of the rotor load by the wing or the /265
reserve of available power render the
ratio
75N
YV
greater than that shown in
Fig. 2. 167 Correction for Rotor
Efficiency as_a Function of ty.
Mo, V.
realization of a large power excess
> 0.45 - 0.5 for a helicopter.
Table 2.13, since then the negative angle
of attack of the rotor becomes greater
than 20° (which is undesirable for design
considerations, since the range of pitch
angles of the helicopter and its parasite
drag will increase) . Furthermore, at
larger
75N
YV
the values of T) may become
smaller than Tl^.p . A tractor propeller or
another propeller may be required for
approximately -~- > 0.35
G
when V„
>
Quantitatively, the change in required power of a helicopter, on installa-
tion of a wing or tractor propeller, is small. Such an estimate will be made in
Section 4, Chapter III.
Section 8. Calculation of Rotor Characteristics inHovering
and Vertical Ascent^ (Mom entu m Theory of Propellers )
The theoiy of a rotor in hovering and vertical ascent has been thoroughly
presented in the literat-ure on helicopter and propeller aerodynamics. In this
Section, we will give some data pertaining to a calculation of rotors with peri-
pheral speeds as they are in use at present.
The calciilations were performed with regard to momentum theory of a rotor.
This theoiy was selected because of the fact that introduction of linearized
aerodynamic characteristics of the profile into the calculation can be replaced
by introduction of the actual dependence of Cy and c^p on a and M, obtained from
wind-tunnel tests of the profile.
1. Brief Review of the M_o mentum Theoir of Propellers
Figure 2.168 shows the velocity polygon in a blade section at a relative
radius r in the regime of vertical climb. The resultant velocity of flow in the
blade section U represents the sum of the vectors: flying speed Vy , peripheral
velocity lur, and induced velocity u. Since the vector of the resultant aerody-
namic force of the section dR is directed opposite to the momentimi vector, the
induced velocity vector u is parallel to dR.
284
J
Fig. 2. 168 Velocity of Polygon in a
Blade Section in Vertical Climb-Regime.
The mass flow through a /266
circular section at a radius r of
width dr is equal to
dm = Q2nrdr\Vi\,
(8.1)
where Vi is the vertical velocity
conponent U.
Applying the momentum theorem
to the ring and using the theorem
of doubling the induced velocity
far aft of the rotor, we obtain the
equation
2dmu=Zf^dR,
(8.2)
where z^dR is the resultant of the elementary aerodynamic forces created by all
blades at radius r.
Substituting eq.(8.l) into eq.(8.2) and expressing dR in terms of the force
coefficient Cr -^q obtain ^^^ equation
4jtq 1 1/, I «r afr = z^^c^^^^^q — b dr.
Equation (8.3) can be represented in the form
(8.3)
|1/Jk = 4-6c„ 6/2.
" 8r sec.
(8.4)
This equation determines the relation between the velocity of the air and
the coefficient of aerodynamic force in the blade section (the so-called
"coupling equation").
Equation (8.4) holds for a stream flow throvigh the rotor disk but is in^
applicable in the region of the "vortex ring" .
Expressing the velocities entering eq.(8.4) in terms of trigonometric func-
tions of the angles of the velocity polygon, we can write eq.(8.4) in the tri-
gonometric form:
sinp-sin(Po — P) of
•=— ^ Co
COS (po — P + v-fr) 8r '^sac.
where p is the inflow angle.
The quantities |j,pr and Pq are equal to
(8.5)
(8.6)
285
%=tarr'-^-. (8.7)
On the left-hand and right-hand sides of the equations, Pq is a known
quantity at a given r; p, ofj., and ^p^ are tmknown. However, if we assign of, ,
then the characteristics of the profile will yield lip, and Cr^ , while
eq.(8.8) will furnish g:
p = a, — 9 = a,-(eo-*ao + A?)- (8.8)
The problem consists in determining a, at which eq.(8.5) is satisfied. This
value of ffj. is found by successive approximations. Simultaneously with deter-
mining Qfj. , we find p, jjbpr, and Cr^
The loads per imit length in the thrust plane and in the plane of rota- /267
tion as well as the torque per unit length are determined from the following
formulas :
^^=bCR^pcos{v-pr-^)s\gnCy; (8.9)
The coefficients of thrust and power of the rotor are determined by numeri-
cal integration of eqs.(8.9) and (8.11).
For an approximate consideration of the tip losses, the loads per lonit
length in the thrust plane are not integrated \sp to the blade tip (r = 1) but xsp
to r = B whereby, according to another paper (Ref .2), we have
B = \-A^ = \-± bo.it. (8.12)
Equation (8.12) can be used when z^ s 3; at z^, = 2, the tip losses should
be taken into account by more accurate methods.
The coning angle and profile power coefficient are fo\md from the expressions
B
Y C dt — .—
I ^ _ _
m
pr-
2. Results of Calculating the Characteristics of a Rotor
The aerodynamic characteristics of a rotor with rectangular twisted blades
286
having a high-speed profile at the tip (-variant II in Table 2-10) for solidity
ratios of the rotor cr = 0.0525; 0.069; O.O9I; 0.11 (the number of blades is, re-
spectively, Zt = 3; 4; 5; 6), k = O.4, -^ — = 1.28*, and for two values of Mq
are shown in Figs. 2.169 and 2.170. Such graphs are used in check calculations
of helicopters in order to determine the rotor thrust in a hovering regime, when
power, flight altitude, and the rotor parameters F, o, u)R are known. The se-
quence of the calculation is as follows: Having calculated nit ^"^ ^
75 AT
75Are;,ge
(8.15)
(8.16)
the graphs can be used for finding the thrust coefficient t and for determining
the rotor thrust
'
-
—
/
7^
/
/•
A
/
/? ?n
nS-
/
\
/
/
.^A
€
^
7^
y
.
^y^;^
/
v%
/
e=75'
--
r
/
/
A
/
/
V
7^
y
n f^
y
~~
\
/
^
'/
r
—
—
k
"J
^^
'7.
—
r
/,
t
..J
-
ie,-o:
p. 00s
0.0m
0.015
Fig. 2.169 Aerodynamic Characteristics
of a Rotor in Hovering Regime (Mq =
= 0.6).
T'—^^i^Rf'^Ft.
(8.17)
The effect of the geometric /268
blade characteristics is illus-
trated ty the graph in Fig. 2.171,
which indicates that the trape-
zoidal blade (variant I) and the
rectangular blade with an increased
twist (variant VI) at t =
= 0.12 - 0.15 require 3 - h% less
power than a rectangular blade with
moderate twist (variant II). Thus,
an increase in geometric blade
twist inproves the rotor charac-
teristics in hovering and in
forward flight (with the exception
of the regime of autorotation) ,
Figure 2.172 shows the radial
distribution of the axial v and
tangential w conponents of induced
velocity. The slope of v and w
with respect to r has a different
character for blades of the ex-
amined shapes. For a rectangular
blade with a geometric twist of 7°
(variant II), v increases from the
root to the tip of the blade; for
a trapezoidal twisted blade and
■55- For other values of k and y» the rotor pitch should be converted by eq.(6.23).
287
Z262
t
-
.
/
rtf
OJO
rl^
'A
4'
^^Y
\^
^
>
r^
1^
p
p^
-
/
/
>
g=/r
>^
/^
0.15
■
/
/
/
p
nz
r
/
/
'.
/.
'V/
'
/.
^.
■
fiw
/.
'/
■;/7;
f
//
//
/A
V/
^•75-
y
0.05
y
J
'm;
.
/
le,=o;
fti7^i"
fli'W
«^7,f
OT,
Fig. 2. 170 Aerodynamic Characteristics of Rotor in
Hovering Regime (Mq = O.?).
o.n
0,10
0.05
-
^
^
^
B
?
^
-
A
y
//
y
A
/
/
y
J
p
/
'-"'6
—
f
0M5
0.010
0.015 m^
Fig. 2. 171 Aerodynamic Characteristics of Rotor with
Blades of Different Shape in Hovering Regime
(a = O.O9I; Mo = 0.7).
288
for a rectangTjlar blade with increased twist, the distribution of v in the tip
portion of the blade is close to \aniform, so that these blades have smaller in-
duced losses. As indicated in Fig. 2. 172, the slipstream velocity w of the heli-
copter rotor is by one order of magnitude less than the axial induced velocity v.
For rectangular blades, the angles of attack of the sections a^ (Fig. 2. 173)
decrease toward the blade tip, and the maximum angles of attack of the sections
are at r= 0.3 -0.5. The trapezoidal twisted blade has a more imiform distri-
bution of Qfp over the outside half of the blade; this angle of attack distribu-
tion, conpared to the rectangular blade, leads to earlier attainment of critical
angles of attack and to a marked increase in c^p at the effective blade portion.
t/
Q.OIO^
0.10
-
(lOS
/f
K
\-w^.
_,
^
.53
i
eT-"
/
■ 1 '
£r
O.O't
O.OC
-\4
O.OOZ
}
y
1
O.Ol
«r
—
-y
n
/
N
J
\
I
r
[^
s^""*^
^
-
^
^>. 1
\,
^
s
\
/270
0,1 P.if 0,6 O.B 1.0 r
0,1 O.if 0.6 O.S 10 r
Fig. 2. 172 Radial Distribution of
Axial V and Tangential w Coiiponents
of Induced Velocity (t = O.I5).
RLg.2.173 Radial Angle of Attack
Distribution of Sections (t = O.I5).
Figure 2.174 shows the dependence of profile power on the thrust coeffi-
cient t and Mq for a blade of the variant II. As we see from the graph, the ef-
fect of air conpressibility becomes appreciable at t > O.I5. The graph in
Fig. 2. 175 shows the effect of the geometric blade characteristics on the profile
power. At Mq = 0.7, the blade profile (variant III; blade without a high-speed
profile at the tip) has the main effect while the blade shape has a smaller ef-
fect. An increase in geometric blade twist (variant Vl) reduces the profile
losses of the blade at large t.
To determine the effect of air conpressibility. Fig. 2.176 shows the graph
of Ztoioo(Mo) = mt(Mo) - mt(Mo = 0.4) for a blade of the variant II; for other
blade shapes, this is shown in Fig. 2. 177. The conpressibility graphs permit
converting the rotor characteristics to other Mq numbers and are also used in
an approximate calculation of the rotor characteristics when mpj. is determined
'by eq.(8.28). We see from Fig. 2. 177 that, for the examined profiles, the blade
without the high-speed profile, at Mq > 0.6, shows a substantially greater inr-
crement in mp^ than the blade with the high-speed profile at the tip.
The values of the thrust coefficients t^p maximally permissible in view of
the flow separation at the rotor blade (see Sect .4*7) were deteiTnined in hover-
ing from the plot of thrust coefficient versus rotor pitch Gq . This dependence
289
* ' -4 'sj^''^^-^^''
/ '^y^^^' ^"^ ^-^
§f V^-^"" ^^
ijy /^,^~ ^
,„ rT1_/^
"' rri /
-4~, / '
U-TV ^^
ttry
uy
gti
jtrT
\L Z
Wl
fi i< . 31 1
4
~t[
'~ ' '
Oi - ^ '^'
tins __ .
/271
t
/
K
ET/
• •
/
.
y^
"
IVl
/r
1
//
/
t
y
m/
/
/ f
/
II
/
t/.^(/
/
II
/
^
/
//
/
/
/
ft/J
1
1
1 1
7
/
-
/
-
1
'
-
-
an^i
0.005
0.010
0,005
0,010 mo
Fig. 2. 174 Profile Power Coefficient as Fig. 2. 175 Profile Power Coefficient as
a Function of Thrust Coefficient t a Fimction of Thrust Coefficient t for
Rotors with Blades of Different Shape
(Mo = 0.7).
and Mq.
6m,
'CO
tm
m
-
1
1
1
/
1
i
<
1
-
/
/
1
X
/
/
/
>
7
^
/
J
f
/
.f
^6.
/
/
/ ^
/
/
y
V
/
y
^
-<
^
c
V-
— ■
■--
"^
-^
^i
0.4
0.5
0.6
0.7 M.
Fig. 2. 176 Increment in Profile Power Coefficient of the
Rotor owing to Air Conpressibility.
290
/272
0.0020
0.00J5
0.0010
0.0005
0.1 Mo
t'-O.ZO
0.5 0.6
m
1
/
n
1
1
1
J
/
/
/
'
/
/
/
/
71
'/^
-^
y
y
0.1 Mo
Fig. 2. 177 Increment in Profile Power Coefficient owing
to Air Conpressibility, for Rotors with Blades
of Different Shape.
0.35 -
0.30
0.15
1
"~---^
~ ==~~"~'^ '"^^
"~" ] — -— ^ ""■•-- ^ v^
"~" ^~- — / — ~- ^ ^ 5
~-~ r^-~-, ^~^i^
^~"?~- ~^^~--, ^^
~'~'~-— ^ "^~~^
■~--~^
"^«
0.H
0.5
0.B
0.7 Mo
Fig. 2. 178 tnax S'S a Fimction of Mq, for Rotors
with Blades of Different Shape.
291
is linear tp to some value of t, after which the linearity is distiirbed; the in-
crease in t with increasing Gq decreases, after which t reaches a maximtmi t^^x*
which is taken as t^^ when constructing the limit of separation (Figs. 2. 119
to 2.121).
Figure 2.178 gives the graph of t„ax ^■s a function of Mq. The diagram
shows that, in hovering, t„ax decreases appreciably with increasing Mq. For a
trapezoidal blade, t„ax is smaller than for a rectangular type. An increase in
geometric blade twist will increase t^ax • The blade without a high-speed pro-
file at the tip has a larger tj,ax » however, as soon as Mq increases the differ-
ence in t„ax will lessen.
According to the momentum theory and with an approximate consideration of
tip losses, the solidity ratio does not affect the angle of attack distribution
over the radius or the magnitude of the coefficients mpj. , Amgo , t„ax» consequent-
ly, the graphs in Figs. 2. 173 - 2.178, constructed for o = O.09I, are valid for
all CT.
3. Approxijiiate Method of Detennining the Dependence /273
of mt on t
For vertical flight regimes of a helicopter, we can obtain an e^qjression
for mt analogous to eq.(3.67) derived in Section 3 for flight with a horizontal
velocity of
I _ _ 1 ^_ _
/n^ = J (div-\-dqw)-{-j c^p bU^ dr~tVy. (8.18)
In this expression, Vy < at a gain in altitude.
Introducing the designations for the terms representing the coefficients
of induced and profile losses of the rotor,
I
mind = ldtv-\-dq'w; (8.I9)
1 _
mpr=-^c^pbU^dP, (8.20)
we can represent the eD^jression for m^ in the form of
"It = m,„d -^-mpr-tVy. ( 8 . 21)
Let us derive the approximate expressions for the conponents of m^ in
hovering .
To determine mimi , let us first assume that the induced velocity v is dis-
tributed Tiniformly over the blade radius and that w = 0.
Multiplying the left and right sides of eq.(8.4) ly cos (|j,pr - p) and using
292
eq.(8.9), we obtain
^■h-dr=dCr. (8.22)
The average induced velocity over the disk v^y is found after integrating
eq.(8.22) with respect to the blade radius from r = to r = B:
^«^=§^- (8.23)
According to eqs'.(8.22) and (8.23), at constant induced velocity the ele-
mentary thrust coefficient is distributed linearly over the blade radius and is
equal to
rfCr = ?^rrf?. (8.2^)
Consequently,
_ To take accoimt of the nonuniformity of axial induced velocity distribution
V and of the term dqw (power losses due to twisting of the flow passing through
the rotor), we will introduce into eq.(8.25) the coefficient i:
'"'"<^==i^^^"'- (8.26)
J2B3
The coefficient $ depends on the planform of the blade and on its geometric
twist, on the solidity ratio, and also on the thrust coefficient. Calculations
have shown that we can take the following average values of I : for a rectangular
blade with a twist of 5 - 9°, a value of $ = 1.05; for a blade with a twist /274
of 12 - 15°, ^ = 1.03. For a trapezoidal blade with taper H = 3 and twist of
5-9°, $ = 1.03. The tip-loss coefficient B for rotors with a = 0.0525 - 0.11
can be taken as equal to 0.98.
Thus, for a rectangular blade with a geometric twist of 5 - 9°, m^ad is de-
termined by the formula
m
tnd
=0.56 V^e'\ (8.27)
The profile loss coefficient of a rotor mp^ is most reliably determined by
the graphs of mp^ = f(t, Mq) (see Figs. 2. 174 and 2.175) which were calculated
for a rotor with similar geometric characteristics. If there are no such
graphs, then mp^ is found from
''P.. „ , (8.28)
293
The first term determines the profile losses at small Mq » while the second
teim takes account of the increment in the profile loss coefficient owing to
• wave drag. An estimate of Amoj, can be made on the basis of the graphs in
Figs. 2.176 and 2.177.
The average profile drag coefficient of
the blade c^p is determined by the profile
polar at the section r = 0.7* for an average
lift coefficient c.
~--'
^
0.005
\^l^
0.00^
Tf'i'
A
y^
Vh
^
0.003
^
"■
O.OOZ
^
•>
■
—
l7,'-t
.01
i
—
•j*
f^
0.001
__
^-
•^
0,10
Ota
o.zo
1
07
-0.001
~-
=
-
^
—
1
«
<?/7/.9
-0.002
*
^
^
"~-^
-
—
^
i>
-0.003
*^
^^
<<?/
?7
>
-
—nnfiL
-0.005
_
_
_
,
_
_
u
'yo
-Zt.
(8.29)
Fig. 2. 179 Graph of the Incre-
ment in Torque Coefficient of
the Rotor in Ascent.
The coefficient P which depends on the
blade shape, its solidity, and on the coef-
ficient t can be taken as approximately
equal to unity for a rectangiilar blade and
for a trapezoidal blade with taper T] = 3,
p = 0.91 (see Table 2.5).
During a vertical ascent or descent,
the magnitude of the induced velocity of the
rotor and thus also mj^^ will vary. There-
fore, to determine mt at Vj_ f 0, it is not
possible to add the term tVy to m^ in hover-
ing flight without considering the variation
in mini . The graph of the increase in
torque coefficient of a rotor during ascent
-0.0k
as
as a function of ?y and t has been con-
0.0it V,
Fig. 2. 180 Graph of Incre-
ment in Rotor Pitch in
Ascent .
Ml
structed for use in approximate calculations.
This graph, shown in Fig. 2. 179, is obtained from
results of calculations made by the momentum
theory. The graph of the pitch increment A9o
dtiring climb is given in Fig. 2. 180.
The approximate ejqjressions for deter- /275
mining Amago and AGq can be obtained from the
following considerations. During an ascent at
low Vy the average induced velocity, according
to the momentum theory (Ref .2), is equal to
(recalling that, with our adopted rule of signs,
V„ < in ascent)
Consequently, in a climb mt and Qq change by an amount of
AW^c = flnd^^ - ntMc/.^ -tVy==t (Vase - %o^) ~ *■ ^y
294
(<-^)av '■a^. 2-0.7 »
or
Aeg=-4iK
y
The aulas o calctilated by the approximation formula is somewhat smaller than
in Fig. 2.179.
Thus, the final e^qjression for an approximate determination of the rotor
torque coefficient as a function of the thrust coefficient and relative vertical
speed has the form
4 . Conversion_gf _Aerodyiiamic Characteristics on Variation
in the Rot or Solidity Ratio
To determine the aerodynamic characteristics of a rotor in hovering flight,
the method of conversion of characteristics can be used. This method should' be
enployed if reliable characteristics of another rotor, close in relative geo-
metric characteristics, are available.
The method of conversion of characteristics in flight regimes with forward
speed, presented in Section 6, is based on a determination of the angle of at-
tack of the rotor at which the velocity polygons, angles of attack of the sec-
tions, and elementary force are retained for a rotor with another a in all blade
sections, i.e., when there is similarity of regimes. In hovering, there are no
similar regimes for rotors with different a ; therefore, the method of conversion
of characteristics is based on the assunption that, at an identical thrust coef-
ficient t, the induced power coefficient in conformity with eq.(8.25) is propor-
tional to — g- and the profile power coefficient is identical.
Thus, if the torque coefficient of the rotor of a solidity ratio a^ - m^
is known, then for a blade of a solidity ratio 03 the coefficient m^ for the
same value of the thrust coefficient is determined by the formula
m*_ = m
t.f="''ind
(8.31)
The values of mp, and B are found from eqs.(8.28) and (8.12).
At an identical value of t for rotors with different o, the angle of 7276
295
attack of the blade sections should be practically the same; consequently, the
rotor pitch varies by a quantity A9o proportional to the difference of the
average induced velocities.
We obtain the formula for determining A9o 5
B _ _ B
^ '
Thus, the pitch of a rotor with a solidity ratio a^ is equal to
60 --=80 4- — l/7(l/Z-)/r,). f^ s
2 I ' 452^^ '-'^ 2 * ^> (8.33)
5 • Determination of Optimal Aerodynamic Parameters of a
Rotor with Consideration of the Dependence of
Characteristics on Mq
To select the optimal parameters of a rotor, it is convenient to use
eq.(8.34) which correlates the rotor thrust, its diameter, and required power
T = (33.25 Y^x\^DNf\ ( 8 .34)
Equation (8.34) includes the relative efficiency of the rotor ^\q, charac-
terizing the relation between the power of an ideal rotor and the actual power
consimied by the rotor^*":
A^ 2m,
t
It is obvious that, at given N, D, and A (i.e., flight altitude), the
greatest thrust of the rotor is achieved at a maxim-urn value of Tlo; therefore,
the designer will strive to approach a maximum TIq. Usually, the graphs of Tig
are constructed as a function of t and a for a value of Mq corresponding to the
average proposed values of the peripheral speed of the rotor (Fig. 2. 181). From
this graph we select a reference point (i.e., values of t, a) with a sufficient-
ly large TIq. We can arbitrarily select the reference point regardless of the
rotor diameter, since an inexact agreement of the value of Mq obtained at the
chosen t, a, D with that for which the gr^h of TIq was constructed, is considered
permissible .
* The power of an ideal rotor is equal to the minimum possible power losses of
a rotor which are directly related with the generation of force, while the actual
power is equal to the sum of all power losses.
296
For a more accurate calculation (with consideration of the effect of Mq on
11 o), the use of graphs T^o = f(t, ct) constructed for different Mq is inconvenient,
since we cannot arbitrarily select a reference point on these graphs. Actually,
eq.(S.15) can be represented in the form
75 N .,3
-^3 - =^,.M„.
(8.36)
Consequently, at -given N, D, H the product nitoMQ should have a definite value,
and on each curve of Tlo corresponding to definite values of a, Mq, only one
point (one nit and, respectively, one t) satisfies eq.(8.36). Two broken curves
are plotted in Fig. 2. 181 for two values of the product m,crM§: the smaller value
of mtoMg pertains to the larger rotor diameter. These curves show that, at /277
given rotor diameter and Mq, only the maximum T]o on the broken curve can be se-
lected and that it is inpossible to realize larger values of T\q . l^on an in-
crease in rotor diameter (lower broken curve) the maximum possible values of Tlo
are still smaller and it is not apparent from the graph whether the rotor thrust
determined by the product TloD increases. Therefore, the graphs of TJo, conr-
structed as a function of t, a, and Mq (or reconstructed as a function of mtaM§,
a, Mq), are not siii table for selecting the optimal parameters of a rotor,
especially if the rotor diameter varies.
Fig. 2. 181 Relative Efficiency of Rotor as a Function of
t and a (Mq = 0.65).
We will find a more convenient form of the graphs for selecting the optimal
parameters of a rotor with consideration of the dependence of the characteris-
tics on Mq . For this, we make use of eqs.(8.36) and (8.37); the latter is ob-
tained from eqs.(8.15) and (8.17):
297
We can show that the ratio
N 75 OTfMo
t
(8.37)
mtMo
on the right-hand side of eq.(8.37) is
proportional to (TIoD)^ . Actually, from eq.(8.36) we determine the rotor dia-
meter D:
const -^ N
D= f. '^
K,)'/2Mf'
and the product TloD:
., n-^ i^")' const /at ,^r-r-r/ t Vl^
^° — o / .1/2 ..3/2 —const VNI- — ^1 ..
It is obvious from this expression, just as from eq.(8.37), that at
given N and H the maximum thrust of the rotor T^^x is achieved at rotor para-
/278
meters at which
mtMo
has a maximtim. Therefore, to find the optimal rotor
parameters we must construct graphs of the dependence of the ratio j^.
am
Fig. 2. 182 Ratio
0.0002
t
mtMc
0.0003 0.0004 0.0005 H!it6M;
as a Function of mtoM^ at o = 0.091.
298
on
nitaM^. These dependencies are shown at a = const in Fig. 2. 182 and at Mq =
= const in Fig. 2. 183 • The maxiimiin values of
these curves.
mtMc
lie on the envelopes of
We see from Figs. 2. 182 and 2.183 that, at given D (i.e., ttligMq = const) and
CT, there exists an optimal value of mJ and at given D and (uR there exists Oopt •
If only D is given, then it follows from these graphs for a number of values of
a and Mq that the rotor thrust increases with increasing a and decreasing wR
(in the range of larger o than that shown in Fig. 2. 183, it may happen that, if
only D is given,' also o^p^ and Moo^t exist). An increase in rotor diameter cor-
responds to a decrease in mtaM| and this, as we see from the graphs, will lead
to an increase in rotor thrust, especially on a decrease in a and Mq .
\
—
-
\
X
^
^
■^
^
s
^
• U 0.10
< D.1Z
' 0.14
1 0.16
n 0.16
0.20
—
V;
!ZS
^
-
—
\
^^
/
ft'""^
—
20
>
-■
-
—
n\^^
<-'/
17.5
/
<N
^
^
^
15
•s.
0.OODI 0.0002
Fig. 2. 183 Ratio
0.0003
0.0004
m^M'o
nitMo
as a Function of
ffltoMo at Mq = 0.65.
All curves are closely spaced near the optim-um of a and Mq, so that any
deviation from optimum values slightly changes the rotor thrust. For exanple.
Fig. 2. 182 shows that, at mtoMo = 0.0003 and ct = O.O9I, we have
19.1 for Mq « 0.55 and tgpt « 0.21, whereas with a 10^ increase in rotor
-'opt
rpm.
i.e., for Mq = 0.6 and t « O.I76 we have
mtMo
= 18.8; consequently, the LZI2.
299
rotor thrust decreases by 1.6^. We find from Fig. 2. 183 that, at the same value
of mtoMo and Mq = 0.65, we have ( — i ^ = 18.65 for o^pt = 0.08, t^pt =
\ mtMo /opt
= 0.165, whereas at a = O.O9I and t = 0.145, we have — = 18. 5, i.e., the
TEtMo
thrust diminishes by 0.8^,
To take account of the variation in the tail rotor losses i^jon a change in
the parameters of the main rotor, the graphs of single-rotor helicopters should
be constructed in the form of the dependence
— --^-^=-- fram m*°Mg (l + C V^a),
rrifil + C y /nfO)Mo
where the coefficient reads
>_ ^fc/ \Dtalt'
C
\Dtaili
I D \3,
The coefficient k^i , which takes into account blanketing of the tail rotor
by the tail boom, is taken to be equal to 1.03 - 1«06.
300
CHAPTER m /280
AEHODYWMIC DESnGN OF A HELICOPTER
Section 1. Basic Equations for Aerodynajni c Design
of a Helicopter
1. Aerodynami c Design Principle of a Helicopter
The principle of aerodynamic design of a helicopter is calculating stable
rectilinear flight regimes in a vertical plane in order to determine engine
power, fuel consvmption, angles of attack, setting angles, and other character-
istics of a rotor during fUght at different speeds and all possible altitudes.
These data permit determining vertical speeds, range, and- dijration under dif-
ferent flying conditions, and also represent necessary material for studying the
equilibriimi conditions of moments (balancing and stability of the helicopter)
and for stress analyses.
Of primary interest is the determination of the performance data of a heli-
copter, i.e., the limiting flight regimes: maximum and minim\jm horizontal flying
speeds at all altitudes, ceiling, maximim rate of climb and range, minim-um power
of horizontal flight, and vertical rate of descent on engine fail\ire.
2. Equation of Motion of a Helicopter
The flight characteristics of a helicopter are determined by solving equa-
tions of stable rectilinear motion of the craft in a vertical plane.
The equation expressing the sum of forces, equated to zero and directed
along the flight path as well as along a normal to it, as shown in Fig. 3.1, has
the form
GsinQfi^i-Q^^^=-X- (1-1)
GcosV^ = r, (1.2)
where
X and Y = conponents of the resiiltant aerodynamic forces of the rotor
directed along the flight path and along a normal to it; at
X < 0, the rotor creates a propulsive force while at X > it
produces drag;
Qp,p = parasite drag of the nonlifting parts of the helicopter;
Qfi.f = angle of flight path of the helicopter to the horizontal.
It follows from Fig. 3.1 and eqs.(l.l) and (1.2) that, in horizontal flight
regimes (9f i.p = O), the Hft of the rotor balances its drag. In flight /281
regimes along an inclined path, the propulsive force of the rotor conpensates
the drag plus the resistance to motion formed by the weight conponent directed
301
II
Fig. 3.1 Forces Acting on a Helicopter
in Steady Rectilinear Flight.
and the power to ^p(u)R)^aF:
along the path, G sin 0f i.p .
To determine the engine power
in different flight regimes and to
find regimes in which the majdmum
power of the engines should be uti-
lized, the equations of motion must
be supplemented by an equation ex-
pressing the condition of equality
of the power absorbed by the rotor
Npg t and the engine power trans-
mitted to the rotor shaft
N^t = Nl.
(1.3)
To reduce eqs.(l.l) - (1.3) to
a dimensionless form, let us refer
the forces to the product ^(u)R)^aF,
1 sin Bf}^-\- c^ — = - t^;
--Q{aR)2aF °
COS QfLp=ty;
YQ(<"^)2<'^
75m
1
-tn,/..
■Q{a>R)3<,F
Equation (1.4) can be then represented as
V
or
iy tanBfip-^- c_^ — - = — t^;
(1.4)
(1.5)
(1.6)
(1.7)
(1.8)
In eq.(1.8) and in what follows, the subscript "h.f ." denotes quantities
referring to horizontal flight.
The given quantities in the aerodynamic calculation are as follows: /2S2
flying weight of the helicopter G;
geometric rotor characteristics (twist, planform), solidity ratio o and
radius R;
peripheral rotor speed coR;
air density p and velocity of sound "a" at design flight altitude;
S c S
drag coefficient of nonlifting parts of the helicopter c^ = — ■:^ — ;
F
engine characteristics: power N = N(H) and hoiorly fuel consunption G^^ =
302
= G(M, H);
engine power utilization factor 5 .
In level fUght (.Qti.-o = 0)» "the number of given quantities is sufficient
for determining the rotor lift coefficient by eq.(1.5). The problem of calcu-
lating level-flLght regimes of a helicopter consists in deterniining, ty means of
eq.(1.7) and for different velocities V, the required coefficient of propulsive
force tjt , finding the values of m* from the aerodynamic rotor character-
istics at known Mq, ty and t,. , and determining \,f from eq.(1.6).
In maximum nonlevel flight regimes, m^ is known (N = N,,^ in c lim bing and
N = in autorotation) : the problem amoiints to determining the values of tx and
ty satisfying eqs.(1.5) and (1.?) from the rotor aerodynamic characteristics at
different flying speeds at known M^ and m^; the flight-path angle 9f i.p is ob-
tained simultaneously.
3 . Various Methods of Determining Ae rodynamic Rotor
Ch aracteristics and_Me thods of Aero dynamic Design
As indicated in Subsection 2, the aerodynamic rotor characteristics, namely
the interrelation of the four dimensionless rotor coefficients V, ty , t„ax, mt
for a range of Mq corresponding to the rpm and flight altitudes of the heli-
copter, should be known in the aerodynamic design.
In certain methods of aerodynamic design, the rotor characteristics are de-
termined by an approximate theory in order to obtain sinple formulas permitting
a direct calculation of the helicopter performance data. Because of the approxi-
mate nature of these calculation methods, they are rarely used at present.
To increase the accuracy of aerodynamic calculations, it is e^^jedient to
separate the problems of determining the aerodynamic rotor characteristics from
those of determining the helicopter performance data. With this approach, the
aerodynamic rotor characteristics can be found beforehand and plotted on special
graphs. This eliminates the need for introducing sinplifications into the calcu-
lation of aerodynamic rotor characteristics. In the Mil'-Yaroshenko method,
presented in Section 2, the following form of graphs is adopted: The angle of
inclination of the resultant aerodynamic force of the rotor to the normal of the
flight path 5, the torque coefficient mt, and the angle of attack a are plotted
as a function of the pitch cp for a series of values of the thrust coefficient t
and the characteristic of the flight regime p, (see Fig. 2. 15).
A further development of the Mil'-Yaroshenko method resulted in a more con-
venient form of the graphs: dependence of the coefficient of p repulsive rotor
force tj on m^ for various values of the lift coefficient ty at V = const and
Mq = const. The graphs also give ciorves of constant values for the rotor angles
of attack by means of which the latter can be defined (the angle of attack /283
must be known for refining the parasite drag of the helicopter and for calculat-
ing a wing-type helicopter or other conposites) .
The graphs for the aerodynamic rotor characteristics can be plotted from
303
experiment or constructed from any rotor theory; the methods of determining the
aerodynamic characteristics are presented in Sections 2, 4, 5, and 6 of Chapt.II.
The method of aerodynamic design in which graphs of the aerodjmamic rotor
characteristics are used, is presented in a very general fonn in Section 3.
It was shown in Section 7 of Chapter II that the aerodynamic rotor charac-
teristics can be determined by using the concepts of performance and propulsive
efficiency factor of the rotor. The method of aerodynamic design based on the
use of these concepts is described in Section 4«
In many methods of aerodynamic design, the e^gjression
derived in Section 3 of Chapter II, is used.
These represent rather sinple but approximate calculation methods. One
such method is described in Section 5.
4. Calculation of Composite and Multirotor Craft
For the aerodynamic design of conjiosite and multirotor craft by the methods
described in Sections 2 and 3, we will construct graphs of the total coefficients
of the lifting and advancing systems of a conposite craft: t^^ (or 6y) as a
function of mty for ty = const. The total coefficients are found experimental-
ly or can be obtained by calculation with respect to known aerodynamic charac-
teristics of isolated elements of the lifting system of the craft. The design
formtilas for determining the total coefficients are given below, for certain
special cases.
These formulas are also used in aerodynamic calculations based on the
methods described in Sections 4 and 5, in which the Hft distribution between
individual elements of the lifting system of the craft must be known. One of
the possible methods of determining the lift distribution between rotors and
wing is given in Section 4« In this case, the formulas derived below are used
for determining the total coefficients of the lifting system of the craft.
Single-rotor helicopter with wing . The simmaiy coefficients ty , t, , and
mt„ are determined by the following expressions (Fig .3. 2):
2-1
t yj, = (^ COS ^arot - ^, sin Aa„^) -j- -^ _ x
X {Cy^ cos Aa^ - c^^ sin Aa^ ),
S 172
'^, = (^xC0SAa^-l-j!ySinAa^f)4 — ^ — X
(1.9)
304
X (% COS ^a^ -f Cy^ sin Aa^ ),
'^ti=filr
or, approximately, Tby
'y.=^ + V
Z28Zt
(1.9')
Figure 3*2 indicates that the angle between the plane of rotation of the
rotor and the path velocity (or velocity of iindistiarbed flow), which we will
call the angle of attack of the helicopter a^ , is equal to
The angle of attack of the wing is correlated with the angle of attack of
the rotor by
a«r =-- a^ — Aa^ + e^^, = a -f Aa^,,, — Aa^ + b^.
(1.10)
In these e^qsressions we denote:
ty, tx, % , a, Cy , Cj^ , (Xk = characteristics of isolated rotor and wing;
Acf, ot = mean downwash angle in the rotor region,
induced ty the wing;
Aff,, = mean downwash angle in the wing region,
induced iiy the rotor;
S„ and ew = area and wing setting angle.
The slipstreams of rotor and wing are denoted by the vectors V^' and V" in
Fig .3. 2. Considering that the flight
velocity is many times greater than
V, the additional vertical velocities of
interference Av, the velocities V and
V are equal to
V" = ]/'V2 + AT);
'%r-
V" = YV'-\-
Af«
V;
V.
(1.11)
Fig .3. 2 Velocities, Angles of Attack,
and Forces of a Single-Rotor Heli-
copter with Wing.
The sequence of determining the
total coefficients at known S^ and e„
is as follows: _
For selected V, Mq, and ty ,
assign Cy and find ty from
the first equation in the sys-
tem (1.9').
From the wing characteristic
305
Cy = f (a„ )y find a,, •
Determine the downwash angles Ao^ot ^^ Affw (see 'below, Subsect.5).
Prom eq.(l.lO), determine the angle of attack of the rotor and from /285
the characteristics of the isolated rotor, find tx and mt .
Note that eq.(1.10) includes the angle of attack of the plane of rota-
tion which, at cpi f^ 0, differs from the equivalent angle of attack (see
Chapt.II, Sect. 2).
Calculate t^^.
After carrying out such calculations for several values of Cy , find the
dependence of tx„ on mt for given values of V, Mq, and ty
y *
Perform
these calciolations for a series of values of ty V, Mq, and then con-
2-1
struct graphs of the aerodynamic characteristics of the lifting system
of the craft.
Fig.3.3 Forces Created by Rotor and Tractor Propeller.
It is obvious that, if the lift of the fuselage (nonlifting conponents) of
the helicopter or a variation in its drag relative to the angles of attack must
be taken into consideration, then the characteristics of the lifting system to-
gether with the fuselage of the helicopter can be determined in the same se-
quence .
Helicopter with tractor propell ers. The additions to the total coefficients
of the lifting system are OD^jressed by the following formulas, which are evident
from Eig.3.3 (interference of the tractor prqjellers with other elements of the
system is disregarded):
A^
A^v
:fc£_
— QO (co/?)2 3l/?2
sin(a + e^^)5K0,
— 0" (<*^)2 nR2
COs(a+e^^);
J^_
CO (<o/?)2 3l/?2
(1.12)
306
75N.
Affix
'V 5
— QO ((0/?)3 ji/?2
In these e^^iressions Pt.p and Wt.p S'^® "the thrust and power of the tractor
propeller, correlated Isy the ratio 75 ^t.^
- Pt.pV
Nt.
When using a cruise jet engine with a thrust of Pt.p on a helicopter, Aty
and Atx will be determined by eqs.(l.l2), and we will have Amt = 0.
Two-rotor helicopter of side -by-side configuration with a wing . For this /2B6
helicopter ty^, t^^, and m^ are determined by the following e^^jressions:
^j.=^(^ ^°^ ^^>r>t-ix sin Aa;.^^) H-^ ^X
■jv_
p
X(Cy cos^a„ —Cj^ sin Aa^),
ix^'='^{i^ cos ^a^,t-\-ty sin Aa^,P +
5^ V^
X
X (Cj. cos Aa^ + Cy . sin Aa^ ),
(1.13)
Unlike eqs.(l.9), here bpi^o^ and Aa„ are the total angles of downwash in^
duced both by mutual interference of the rotors and interference between rotors
and wing.
The sequence of calculating the total coefficients is the same as for a
single- rotor helicopter.
Equations (1.12) are added to eqs.(l.l3) if tractor propellers are present.
Two-rotor helicopter of fore-and-aft configuration . Disregarding downwash
in the region of the front rotor caused by the tail rotor, we can obtain the
following relations ( Fig .3 '4) :
>!)"
--iy^-\-i.iy, COS Aa„f. — t^, sin ^ar,u) « ty^ -{-ty„'
^x^ = ix, + {tx ,Q.os Attnt, + iy, sin AOnt.) ^ ^x, +
+ {ix, + ty,Aa„t,),
02 = Oi — Aa;„f, + ASrete ^
(1.14)
307
In these expressions, we denote:
Affrot ~ mean downwash angle in the region of the tail rotor, induced
by the front rotor;
= effective angle of advance of the tail rotor relative to the
front rotor (with consideration of the difference in deflection
of the automatic rotor pitch control):
Ae
r t^
A^iwfj = Ae^j/-
-(D,x),+(D,x),.
(1.15)
The subscript "1" denotes characteristics of the front rotor and the /287
subscript "2", of the tail rotor.
The performance data of a heli-
copter of fore-and-aft configuration
can be uniquely determined if and only
if the conditions of joint operation of
front and tail rotors are known. Usual-
ly, such a condition is the • relation
between the thrust of the rotors deter-
mined by longitudinal balancing of the
helicopter. Knowing this relation for
selected V, Mq, and ty , it is possible
2-1
to find tv, and t„ from the first
equation of the system (1.14). After
assigning ffi and calculating Aofpotp* we
Fig. 3 .4 Velocities, Angles of
Attack, and Forces for a Heli-
copter of Side-by-Side Configura-
tion.
can determine a^ • From the characteris-
tics of the isolated rotor, knowing V,
0, ty, and a we then find t^ and mt for both rotors. Furthermore, t^v and mt„
are calculated and graphs of the aerodynamic characteristics of the lifting
system are plotted.
M.
5. induction Coefficients of Two-Rotor Helicopters
and Helicopters with a Wing
Determination of the lift and drag of the system of lifting elements is a
conplex problem for whose solution the induced velocity and loads per unit
length in each section of the lifting elements should be found, with considera-
tion of the effect of all vortices entering the system. When using high-speed
conputers, solution of this problem is possible in certain cases.
However, usually in aerodynamic designing, the confutation is limited to
deteniiining the average downwash angles Aa of each of the elements of the lifting
system. As shown in Subsection 4, the downwash angles permit finding the pro-
jection of forces of all elements of the lifting system onto the direction of
motion and normal to it.
Equations (1.9) - (1.13) show that, for Aff> (i.e., for the vertical in-
duced velocity conponent caused by other elements of the lifting system, Av is
directed from the top downward), the drag of the craft increases by an amount
308
/288
I'lg.3.5 Induced Velocity Distribution of Wing
(Points 1, 2, 3, 1', 1' , y are above and below
the Vortex Sheet).
J/
■'.3
'
1 1 1
y=''<'Jl
rl K=^
\\
\
■1,0
^
fl/^
7TN
\
' 1
<.\
' 0.1
\
tV
j /
s
is
t o-J~
^
h
V .•^>
•^
<^
=^
5^
Ui^
p
^
^
=
■^
si
?^=o
SI
1/ ^'-^
r"
= ^E
p
—
0,5
Viii^
T
/.ff
1
10
^z
,-3>aj ^
V
~^£_^— ■ ^^
'^ ^
l^-^l-^'-;
'2^
0.5^
>yT
"nfr-^ t^^
4S
-ZZ
"" ^ ^
-' /
"•' .--^'
^
in-
/-^
Fig. 3. 6 Induced Velocity Distribution along Wing Span z
at Different Distances from Vortex Sheet y.
309
YAqt while, at H^a < 0, the drag of the craft decreases.
The downwash angle of the i-th element of the lifting system induced by the
n-th element is determined by the expression
Aa, = ^, (1.16)
where AVj is the vertical induced velocity conponent created by the n-th element
at the focus of the i-th element, averaged over the area of the i-th element;
Avj is proportional to the mean induced velocity in the plane of rotation of the
n-th element:
Av^ = yi,v^^ =^i — ^. (1.17)
n ' 4B2V
The proportionality factor n^ is called the induction coefficient. This
depends on the mutual arrangement and dimensions of the i-th and n-'th elements
of the lifting system.
Let us recall how the induced velocity of the wing (or rotor) is distributed
in space. At points downstream of the wing, the induced velocity increases and
rather rapidly reaches double its initial value (Fig .3.5). An increase in in-
duced rotor velocity will then take place within the rotor disk (Fig. 2. 3).
At points upstream of the wing, the induced velocity is virtually equal to
zero, while it decreases at points above or below the vortex sheet (points 1,
l', 2, 2', 3, 3' in Fig. 3. 5). In cross section, the induced velocity of the /289
wing with an elliptic circulation distribution has the form shown in Fig.3.6:
Within the span of the wing or rotor ("i = — - — 7— < 1.0 j, the induced velocity is
directed downward while at the periphery ("z s l.O), it is directed ijpward.
let us determine the magnitude of the induction coefficients.
The coefficients of mutual induction depend on the flight velocity (V) and
on the angle of attack of the rotors; otir values of h cire averaged with respect
to V, approximate, and applicable to all flight regimes at V s: 0.15.
The rotors of a helicopter of side-by-side configuration, as is obvious
from Fig.3.6, are located in the region where the induced velocities caused by
the adjacent rotor are directed from the bottom vp. In this configiiration, the
interference reduces the induced drag of the system. Here, the value of the
mutual induction coefficients Hs . g was taken from B.N.Yur'yev's book (Ref.2) to
which corrections were applied for the fact that the induced velocities at azi-
muth t|f = 90° are greater than at azimuth ^ = 27CP . Therefore, the mutual induc-
tion coefficients depend on the direction of rotation of the rotors: When the
azimuth i|f = 90° is between the rotors, these coefficients are approximately 25%
higher than the mean values obtained elsewhere (Ref.2), while they are about 25%
lower in another direction of rotation.
310
*SJ
(.
-o,s
^
\
/
\
s
1
-ft*
r
\
\oin,l.
\
-0.3
^
N
\
f
\
\,
V
V.
-o.z
s
V
^v
-II.I
^.^
n
Wrjl
I.S
1.0
Fig. 3. 7 Coefficient of Mutual Induc-
tion of Rotors for Helicopters of
Side-by-Side Configuration.
1.0
0.9
0.8
0.7
0.6
0.5
\
\
\
t-?^^
\
ir
. Jf[^ '
— N
\
^Lt^^ ♦
\
V
\
X
^
^
0.1 o.z 0.3 O.it 0.5 y
Fig. 3. 8 Coefficient of Mutual Induc-
tion of Rotors for Helicopters of
Coaxial and Fore-and-Aft
Configuration.
The graph of Jtg . g as a f iinction of the distance between the rotor axes
z' = — ^ is shown in Fig.3«7« It is obvious that the optimum distance between
R
rotors, at which the least induced drag in forward flight occurs, is equal to
z « 1.8.
If, in helicopters of coaxial configuration, there is no vertical separa-
tion of the rotors, then the coefficient of mutual induction k^o obviously will
be equal to unity (Av = Vav)« When there is vertical separation, the induced
velocity in the plane of the second rotor Av will decrease (Av < Vav ) so that
Kgo < 1. The graph of k^o as a function of the vertical separation of the
rotors y = -^, taken from another paper (Ref .2), is shown in Fig. 3. 8.
R
According to the general theory of induction, the mean induced velocity of
the system of lifting elements does not depend on their stagger in the direc- /290
tion of path velocity; consequently, for a helicopter of fore-and-aft configura-
tion the mean magnitude of the additional induced velocity Av is the same as for
a coaxial helicopter (at equal y) . Since the tail rotor does not influence the
front rotor, we have K^ot =0; consequently, for the tail rotor located behind
2AVa
so that H
r otj
= 2>l.
Thus, the mutual induction
the front rotor, Av
coefficients in helicopters of fore-and-aft configuration are also determined
in accordance with Fig. 3. 8.
In terms of the general induction theory, a decrease in induced drag for
^«- In Chapter I, in eq.(3.22), we had h = 2k^
311
•t*^
ysO
Side-by~side
configuration
helicopters of side-by-side configtiration
(hs . , > O) and an increase in induced drag
for helicopters of fore-and-aft and co-
axial configiirations (hoo > O) is e^qslained
in the following manner: It is known that
the induced drag of the system of lifting
elements is directly proportional to the
square of lift and inversely proportional
to the mass of air participating in produc-
ing lift, or to the effective cross section
of the air stream (see Fig. 2. 56). In the
ideal case (uniform induced velocity dis-
tribution over the entire span), the effec-
tive cross section of the air stream is
equal to a circumference whose diameter is
equal to the span of the lifting system.
Fore-and- aft
' confieuration
Fig.3.9 Effective Cross Section
of Air Stream for Helicopters of
Side-by- Side and Fore-and-Aft
Configurations .
The effective stream cross section F^
for helicopters of side-ty-side and fore-
and-aft configurations is given in Fig. 3 .9 •
The sketch shows that, for the side-by-
side helicopter, F, is greater than the
area of the two rotors ( Kg , g < O); at
"zs.s > 2, a gap effect appears and F^ de-
creases. In the fore-and-aft configuration
without vertical separation of the rotors (y = O), the effective stream cross
section is the same as for a single rotor (.k^o =1); ^^ the presence of vertical
separation, Fg increases (koo < !)•
Now let us examine the interference between rotor and wing for single-rotor
and fore-and-aft helicopters (Figs. 3. 10 and 3*11) • It is obvious that, if the
wing of a single- rotor helicopter is very close to the rotor (y = O) and the
spans of both rotor and wing are equal f Ivt = p - = 2.0J, then k„ « K^ot " I'O.
l^on an increase in l„, due to the fact that the induced velocity is directed
upward outside the rotor, the induced velocity of the rotor averaged over the
wing span will decrease (h„ < 1»0), while the induced velocity of the wing
averaged over the rotor area will change little. Correspondingly, ipon a de-
crease in ly, , K„ will change little whereas K^ot will decrease. At T„ < 1.0,
when the wing is underneath the ineffective blade sections, h„ will decrease.
The graphs in Figs .3*10 and 3«11 are valid for a helicopter of fore-and-aft con-
figuration, but must be taken with consideration of the mutual longitudinal dis-
placement of both rotors and wings; for elements located aft, h is doubled, /291
whereas for elements located forward, h decreases to zero.
For a helicopter of side-by-side configuration, let us examine one of the
rotors in calculating k. At "z = 2.0, half of the wing is underneath the rotor
and half is outside the rotor (Figs.3«l2 and 3 •13); therefore, h„ and H^ot ^^®
smaller than 0.5 (they would be equal to 0.5* if the induced velocity were equal
to zero outside the rotor disk or wing span and were uniformly distributed with-
in their confines). I^on a decrease in z, all larger conponents of the wing
312
l.O
o.s
__
^
/
/
/
/
1.0
i.O I,
'Rotor
-^^
'if>=270'
vwo-
2.0
u >z.o
Fig .3. 10 Induction Coefficient of Rotor for a Wing on
Helicopters of Single-Rotor and Fore-and-Aft
Configurations •
and rotor will be within the field of induced downward velocities, so that both
K^ and Hyot will increase.
The nvunerical values of h, and n^a\ plotted in the graphs (Figs.3-10-3»13)
are given for y = 0. A decrease in k at y ^ can be determined from the graph
of Keo = f(y) in Fig.3.8, i.e..
''/(!/l) = '</(!/ = 0)''co(i/l)-
(1.18)
Thus, in accordance with eqs.(l.l6) and (1.1?), the downwash angles are de-
termined by the fonniilas
Cy Cy iy o
Aa,=x, — ==-,=0.26x, =^=0.26x, -^.
(1.19)
The downwash angle of the rotor induced by the wing can also be found from
the ea^ression
Aa„* — t-rot "7 — — '*-rot 72 77;
aX
ll^v^'
(1.20)
313
m2
Xrot
i.U
, ,
^
^
/
/
0.S
/
t
r
/
10
10 3.0 L
-Wing
■Rotor
l^v <z
Fig, 3. 11 Induction Coefficient of Wing for a Rotor on
Helicopters of Single-Rotor and Fore-and-Aft
Configiorations
o.s
X
X
*^
V
Oir.fl.
' —
—-
■5— '
\
"V,
~~' ~
—
CLs
1^
Dirfl.
@i
US
2.0
^te|^
0.3
O.l
0.1
■s
S
^
_
-
—
-
—
"
uo
1.5
2.01
Fig.3.12 Induction Coefficient of Rotor Fig.3.13 Induction Coefficient of
for a Wing on Tandem Helicopter. Wing for Rotor on Tandem Helicopter.
31ff
Section 2. Aerodynamic Helicopter Design by the /293
Mil^-Yaroshenko Method
Let us examine steady regimes of rectilinear motion of a helicopter with
low flight-path angles to the horizontal.
Assuming the thrust of the rotor to be approximately equal to the weight
and considering the revolutions of the rotor to be given, flight should always
take place at a constant thrust coefficient t. In this case, the magnitude of
the projection of the resultant onto the direction of motion can be varied only
after having changed the angle of attack; at the same time, also the rotor pitch
must be changed and hence the power transmitted to the rotor, so as to maintain
balance of forces with respect to the vertical.
The method proposed below for designing a helicopter assimies, for each pos-
sible value of rotor pitch, that the aerodynamic rotor characteristics (thrust,
longitudinal force, and torsion) aire known.
1. Equations of Moti on and Design Principle
Figure 3-14 shows the forces acting on a helicopter in steady rectilinear
motion .
G
Horizontal flight
Fig. 3. 14 Forces Acting on a Helicopter in
Steady Rectilinear Motion.
The equations of motion of a helicopter can be written in the form
} (2.1)
The angle between the direction of the resultant and the nonnal to the path
/?sin8 + Osine/i^-|-Q^,= 0;
;?cos8 — Gcoseyi^=0.
reads
8 = a+ /a;,-/ y- = a +/■«/?-' y-.. (2.2)
315
0.0W
0.005 -I
mk
0^-t.O
Fig .3. 15 Aerodynamic Characteristics of Rotor
(p, = 0.15; t = 0.13; a = O.065).
-2
-4
-5
-8
-«?
-n
-%'-'
:_i,
1
=
^
;?
«^
y
innn_
„^.
^'rf.-a ^
M'^ZOOOm
^
—
r-
r ■= —
~"~
rc;^
1 H=mOm
-0.1
•^
^
_^
^
yiLIii— •
-
■~,
-
<>'</
te \
'^
-—
'\s
-0.2
S
__^
1 ■
[ ."^
?3
^
"^
•i
•^
^=20on^
otrfte •
■^
m
ffffl
-^
—
—
—
V
H»0
.^
-'
•n "
N,
^
^-
'■'
■■
^
-J
ftta
0.20
a3o (U
Fig. 3 '16 Required and Disposable Characteristics
of Helicopter.
316
Below, we will assume that the angles 6 and 9fi.p are small. Fiorthermore,
owing to the smallness of -^ in flight regimes, we can assume the resultant
force of the rotor as equal to the thrust (R = T). Then eqs.(2.1) can be re-
written in the form
T=Q. \ ^'^•^''
The angle of inclination of the forward resultant 61, . ^ required for hori- /295
zontal flight is found from eqs.(2.3), setting 9f i.p = 0:'
^V=-^- (2.4)
The flight-path angle for any given regime will be determined then from
eqs.(2.3):
%i.p=~{^-h.f)=~l^'-
(2.5)
Thus, the problem consists in determining the possible angles of inclina-
tion of the resultant 6 for each given regime.
Figure 3.15 shows the angle 6, the angle of attack or, and the torque coef-
ficient of the rotor m^ relative to the condition of constancy of the thrust
coefficient t, as a function of the blade pitch cp for a specific flight regime \i,.
The larger the setting angle cp, the more negative must be the angle of attack a
of the entire helicopter, so as to maintain balance of forces with respect to the
vertical, and the larger must be the angle of inclination of the forward resultant .
The graphs in Fig. 3. 15 show that large setting angles often require a larger
torque, i.e., a greater expenditure of power. Hence it is clear that, after de-
termining the magnitude of the torque, it is possible - for exanple, from the
total engine power as shown in Fig. 3. 15 - to obtain the maximum (disposable)
pitch cp4ig for a given regime and hence the corresponding magnitudes of the
angles of inclination of the resultant 641, and angle of attack a^is . Converse-
ly, on assigning the value of 6 - for exanple, from the condition of horizontal
flight - by means of eq.(2.4), it is possible to obtain the required blade set-
ting angle cp, the torque coefficient mt, and the angle of attack of the rotor a.
Figure 3. 16 gives the resiJ-tant values of 641s and 6^,f as a function of \i..
The graphs in Fig. 3. 16 are also the main graphs for the calculation, by means of
which all necessary flight data can be determined. The intercept of the ciirves
determines
The vertical velocities as a function of (j, or V can be found from the
formula
(2.6)
317
where the quantity 9fi.p = -A5 is taken from the gr^h in Fig. 3 .16.
2.. Determination of- Aerodynamic Roto r C harac teristics
The quantities mt, 6, and cv as a function of the pitch 9 and at a given
value of the thrust coefficient t - for exanjjle, similar to those shown in
Eig.3.15 - which are necessary for calculation, can be determined e^qjerimentally
or theoretically.
Within certain limits, the Glauert-Lock theory gives results close to
reality (Sect. 2, Chapt.Il). These limits are boiinded by a certain regime p, and
by the magnitude of the thrust coefficient t, characterizing the value of the
average working lift coefficient of the blade section Cy and thus determining
the admissibility of ass'unptions made in the theory for the linear dependence
of Cy on the angle of attack and for the possibility of adopting an average /296
value of the coefficient of profile
h'
T '"1
0^0
0.10
0.005
^
1 1
1
(/)=0
■25-
-
-
1
-
-
\
^
-—
—
0^
J
—
<
^
^
^
X
.
^
0.
^
"=
■—
.^
--
r~~
(7i
-N^
k
<.
"
mt
—
'^•1^4^
^
N
k,
\
^^v
r
N
X
2
\
N
S
*
S
-^
^
-0.005
X
^
"
\
'
^
\
^
^
■—
-^
^
^
<^
\
\
%
^
'
;>
y'
\
L>
■^
^
>
^
\
^
^
■<
c
0.
'0.
^
•^
^
\
t
ndfi
^
^
<*
■1
-aoi
1-
L
_^
u
_
U
L
o.w
0.20
0.30
drag Cxp = const which does not depend
a V
on the angle of attack of the section.
In the aerodynamic design of a
helicopter it is convenient to define
the flow coefficient X on the basis of
eq.(2.50) for the thrust coefficient
(see •Chapt.Il). For this, we make use
of the second equation of the above
system (2.3) which 0:153 resses the condi-
tion that, for any rectilinear motion,
the rotor thrust is approximately equal
to the weight of the craft. Thus., the
thrust coefficient in helicopter flight
at a given rpm in all rectilinear re-
gimes is constant, and its value is de-
termined from the condition T = G. Then
X, at given values of n and cp, will be
determined from the equation
Fig. 3.17 Auxiliary Graphs for
Calculation of Rotor Characteris-
tics (p. = 0.1).
X=
2i
a«,B'i
■Ht^+4)- f^-^'
If now we substitute the value of \
into the ejqjression for h (2.68), m* (2.47), and o- (2.53) in Chapter II and plot
their dependence on cp (see Fig. 3. 15), then each point of these curves will cor-
respond to one of the possible regimes of rectilinear flight.
To sinplify the calculations, let us plot graphs of the quantities
ft — -^ /•--
h'
*-!" ''"a.
t
t
(2.9)
3I8
and
ntt
4a„
(1 + f^^)
as a function of t, where cp is a parameter.
Figiires 3.17 - 3.20 show these graphs, plotted on the assijnption that /297
the coefficient of tip losses is B = 1 and that the mass characteristic of the
blade is y = 5. On a change in flight altitude, y will vary in direct propor-
tion to the variation in air density. As a consequence of a variation in y ,
also -^ — and m^ will vary, but the changes in these quantities are small for
values (i, ^ 0.3.
0.20[a005
T ^i
0.20
0.10
0.
70?
~n
..
—"
?
JO
^
--
^
^
1
,^
a
5>
-=
fc=
'^
Sfc
^
V
—
—
v«
777'-
>»
^
^
■v
L c
3 -
■0
^v
N
N
^
s
s
s
-0.005
V
N
V
\,
\
k
>
\%^
•»..
-
—
—
—
;s
r^
^
%
j>
—
s
—
— 1
=
=
^
^
K
-0.010'^
• /
"
-'
0.
^
—
^
5
t
'NM
_-j
-^
^
—
<f
'0.0^
-^
"
-
1015
1
V.W
0.Z0
0.30
Fig. 3. 18 Auxiliary Graphs for Calcula^ Fig .3. 19 Auxiliary Graphs for Calcu-
tion of Rotor Characteristics lation of Rotor Characteristics
(|j, = 0.15). (iJ- = 0.2).
The quantities m^ and
represent conponents of the coefficients of /298
torque and longitudinal force due only to lift and induced drag of the blades;
the conponents of these coefficients due to the profile drag of the- sections do
not enter into eqs.(2.9). For values of |j, within limits from 0.1 to 0.3 and for
the usual profile surface finish, a value of c^p
results •
= 0.012 gives satisfactory
319
■■■■■■■III iiiiiiii II inn
llllllllllllll
llllllllllllli nil ■IIIIIIII
3. Calculation of Flight Data
A selection of basic parameters usually precedes the aerodynamic design.
Let us assume the rotor diameter as given.
Obtainment of optimum flight data in vertical regimes requires minimim loads
on the disk area; therefore, the rotor
^, diameter is selected as large as pos-
sible with respect to design and weight
considerations. Also the magnitude of
the solidity ratio a is mostly predicated
on design considerations.
0.20
0.10
Ws
,
\^
3s
ilJ f"'^"^
0,005 \ 1
>U w=0
iO^^^^K^y^'A
--^^^z-^
--'"N--^:;^^
"^ x^'^^K^^"^
\ ^)9>>
'^^^S'^/^K
^-^ ^x^^s^ s^
_\£P V_-'^
^^^^^ N^
^Zc^^^ i:
-0.005 V^ =^
^^^'^.^N N
IK -^ y
2- '"* "^^ ^^
J ^^ -^
^. \
-^^■w ^^ >^
^ ^
^
-wfo o^
. ^ ^
^
-0,0 s
0.10
0.20
0.30
Fig. 3. 20 Auxiliary Graphs for
Calculation of Rotor Characteris-
tics (p. = 0.3) •
The magnitude of blade loading
which determines the working Cy of the
section thus depends largely on the
rotor rpm. For a rational selection of
the rpm it is therefore suggested to
assign 3 or 4 values of the angular
rotor velocity and to perform a conplete
calculation for these.
As regards available power trans-
mitted to the rotor, in the case of two-
or multirotor configurations it is
necessary to account for the efficiency
of transmission and for losses due to
cooling; in the case of a single-rotor
configuration, the power esqjended for
driving the tail rotor must also be taken
into account. In first approximation,
this power can be found for hovering
flight and is taken as unchanged in
forward flight, which will yield smaller
values for the performance data in /299
forward flight than can be e^qjected in reality.
The sequence of calculation is as follows: After assigning several values
of (ju, a series of values of t is derived; for given values of ^x and cp, the rotor
angle of attack a and the coefficients h and mt are determined, and eq.(2.2) is
used for defining the corresponding values of 6 .
The found values of 6 and mt are plotted as a fimction of cp in the form of
graphs similar to those in Fig .3. 15, each of which is constructed for a definite
value of |J,. Then, plotting on the y-axis the values of the available torque
coefficient
m^
'dis
(2.10)
the corresponding values of disposable 6413, ofiis, ^.nd cpaig are obtained. The
320
I l^ll I nil II I IH
■ IIIIIHIIHMIIIIIIIII nil I llllll IIIHIIIIII nil II
5
3
Z
^y max
r^
"^
^.
n=Z00rpm ^
\
1 1
-
■ts.
N
\
n=Z'iO
y
\.
N
V
V
\
--
s
\
•
\
N
77=285
'^
"N
\
\
S
>
A
>(
50
\Ih
700N
\.
\
f?
>
>
■
v\
Vkmthr
Fig. 3. 21 Rate of Cllirib of Helicopter
as a Function of Flight Velocity.
next step is to determine the
values of e^.f, Q?h.f> 9h.f* and
b. f
required for horizontal
flight and to construct a graph of
these values plotted against |j,
(see Fig .3. 16 J in the same manner
as presented atove.
The power required for hori-
zontal flight N^.f is found in
terms of the torque coefficient
mt by means of eq.(2.10).
Having determined, by
eq.(2.7), the values of Vy as a
function of. V, we then construct
the graphs shown in Fig. 3.21.
From these graphs, we find the
values of V,
and V„
and the
optimim rate of clLmb V,, for each flight altitude and rotor rpm. Data corre-
sponding to other altitudes can be determined in the same manner as that given
above; the graphs, shown in Fig.3.15, should be constructed for values of t^
corresponding to a certain height on the basis of the relation
Qo
<o
/ f «0 •■0
To obtain fUght data for a helicopter with respect to height above ground,
it is also possible to use the
following method which does not
require constructing the graphs
shown in Fig. 3. 16. A change to
another altitude is characterized
by a change in p . The graphs /300
will remain unchanged if the value
of the thrust coefficient is re-
tained . Since
Fig. 3. 22
150 Vkm/hr
t =
nR2 (u/?)2
Graph of Helicopter Flight
Data ( n = 240 rpm) .
then, for constancy of t, we must
retain the equality Pooul = PhU^h •
Hence, we determine the value of
oJh at which t and thus also all other coefficients remain constant. The curve
of 6i,.f = f(iJ.) in Fig. 3. 16 remains the same, since the drag Qp^^ depends on piw^ ,
and this product does not change with height. The available torque coefficient
must be • calculated for power at an altitude, with consideration of the new value
of angular velocity. If the power at altitude N^ = ANq, then
321
Ha-ving (mt )h > the described process of obtaining 6^18, cp^iB* and ot^is is
repeated; the resultant values are plotted on graphs as shown in Pig .3. 16.
After -determining A6, graphs of Vy = f(V) are plotted, finding ¥„ and V,
a function of rotor rpm for various heights taking for each height n^ = Hq
max as
1
Va
These calculations must be carried out for at least three values of rotor rpm.
Then, recording from the graphs the values Vy and V,, ^x with respect to /301
ID & X
altitudes for given revolutions, a graph as shown in Fig. 3. 22 is plotted, from
which we can determine the ceiling (dynamic) and also the variation in Vj, ^x with
altitudes, at given revolutions.
Vc
--
5
>J
u
-
\
s,
't
\
\^
\
\
/
3
\
s\
5<
/U =
U.I
k
k
••
s
n
%
s
2
\
^
41
1
He rpm
250
200
— <ft=3'
H^zom
woo
2000
^V-r-t-^ TCi-J
150 Vkm//ir
ZOO
Z50
n^rpm
Fig. 3. 23 Rotor Pitch in Autorotation Fig. 3. 24 Rotor Rpm in Autorotation
Regime . Regime .
The calculations for any weight can be made just as for any altitude, i.e.,
using the graphs shown in Fig. 3. 15 and changing only the value of n in conformi-
ty with the variation in weight.
From the condition mt = 0, the pitch cp for an autorotation regime is deter-
mined. After constructing graphs (Fig. 3. 23) of the dependence of cp^ on the num-
ber of revolutions of the rotor m^, for each value of m., we can find the rotor
rpm in an autorotation regime as a -function of flying speed and for any constant
value of cpo . The dependences n^ = f(V) for different flight altitudes are
plotted in Fig. 3. 24.
4. limits of Applicability of the Method
The presented method permits analyzing the influence of numerous parameters
322
that determine the flight regime of a helicopter. Moreover, the degree of accu-
racy of the calculation of performance data, based on this method, is fully de-
termined ty the extent to which the theory underlying the calculation of rotor
aerodynamics yields results close to reality.
In regimes |j, -within the limits from 0.1 to 0.3 and with thrust coefficients
smaller than the majcmum permissible, the section angles of attack are within
the range for which the assimptions made in the theoiy are valid (cy = 3.^0,.;
c-r, = const).
At large values of \i and, in particular, at large blade setting angles,
the section angles of attack in a large portion of the disk area exceed the
critical value, and flow separation takes place. An ultimate analysis indicates
that the theory in these regimes gives values of the longitudinal force, and
especially of the torque, that are lower than reality, and also produces errors
in the angle of attack. Thus, the results of the calculation by the proposed
method should give higher values of maximum speed if this is determined in the
region ij, > 0.3- The assimption of a uniform induced velocity distribution does
not hold at small |j, (|j, < O.I5). In reality, the induced losses are larger in
these regimes owing to nonuniform induced velocity distribution so that the
calculation will give larger values of the rate of climb of the craft.
These errors are small (of the order of 10^) for helicopters with low disk
area loading, but markedly increase with increasing G/F, i.e., with increasing
relative percentage of induced losses.
An increase in the accuracy of calculation of the flight data can be
achieved by refining the theory or by using data obtained from wind-tunnel tests.
Section 3 • General Method of Aerod ynamic Design for
Rotary Wing Aircraft
In this method of calculation, just as in the Mil'-Yaroshenko method, the
first step is to plot - on special graphs - the aerodynamic rotor characteristics.
Then, the propulsive .force coefficient t^ is plotted against the torque coeffi-
cient mt, for constant values of the coefficients ty, Mfi (or V), Mq (see
Figs. 2. 142 - 2.145). To calculate a helicopter with a combined lifting system,
the same graphs are plotted for the total coefficients; the design formulas for
determining the total coefficients were derived in Section I.4. Thus, this 7302
method of calculation encompasses all types of rotary wing aircraft.
In determining the perforaiance, i.e., in solving the equations of motion,
no sinplifying assuirptions are made and the accuracy of the calculation is de-
termined by the accuracy of the graphs of aerodynamic characteristics of the
lifting system and by the correctness of estimating the parasite drag of the
helicopter and the engine power losses. We make only the assunption that the
performance data can be calculated separately from the balance calciilation at
some average (for a given centering of the helicopter) value of deflection of the
automatic pitch control Kav • This leads to an error in determining the angle of
attack of the fuselage and wing; therefore, at great differences between k and
Hav for a helicopter with a large wing (S„/F > 0.05 - 0.0?), the assunption is
323
no longer valid.
A shortcoming of this method is its relatively great e^^jenditure of time.
Consequently, it ranges among methods of final aerodynamic design. However,
whenever graphs of the aerxjdynamic characteristics of the lifting system are
available, the calculation of the performance data is not excessively laborious
and the method can be used also for preliminary calculations.
In the calculation, aiaxLliary graphs suitable for all craft with similar
lifting systems and equal parasite drag coefficients are constructed. By means
of these graphs, plotted once and for all, numerous aerodynamic design calcula-
tions of versions of a craft can be performed, including calculations for dif-
ferent conditions of helicopter use (variations in flying weight, rotor rpm, or
atmospheric conditions).
1. Construction of Auxiliary Graphs for He li-copter
Performance Data
In this Subsection, we present a method of constnicting auxiliary graphs
for calculating helicopter perfxjrmance data. Strictly speaking, these graphs,
constructed for a helicopter with a specific lifting system and specific de-
pendence Ck on oif f are applicable only to this type or to other helicopters with
similar lifting systems and identical dependences of c'^ on at • However, the
graphs can be used with sufficient accuracy for all helicopters of the same con-
figuration having identical values of o, c', and other dimensionless charac-
teristics (for exanple, ew> S„ for a helicopter with wing) and Mq not greater
than 0.55 - 0.6, when the blade shape does not excessively influence the aero-
dynamic rotor characteristics. Therefore, at Mq ^ 0.55 - 0.6, the aiaxiliary
graphs are universal. Characterizing the parasite drag of the helicopter only
by its magnitude at cif = 0, "c^, _ , it will be assumed that the increment of "Cy.
upon a variation in c^f can be considered identical for helicopters of the same
configuration. ¥e will disregard the lifting force of the nonlifting elements.
For helicopters with a narrow variation range of Mq (flight at constant
rotor rpm; dynamic ceiling less than 5000 - 6000 m) and with a maxunim Mq less
than 0.6 - 0.65, the aioxiliary graphs are constructed for a mean value of Mq.
For helicopters with higher Mg, determination of the performance data for the
mean value of Mj, leads to noticeable errors, as a result of which the auxiliary
graphs lose their universality and can be used only for one value of Mq.
The method of enploying the graphs for determining performance data of a
helicopter is presented in Subsection 2.
Auxiliary graphs for required helicopter po wer. In horizontal flight /303
of a rotary wing aircraft (9f i.p = 0), the equations of motion (1.6) and (1.7)
take the form
/2
c
^= -/...: (3.1)
•« - ■"■h.f
32U
I/2QOIt;?Z(o,/?)2 y% • (3.2)
In eqs.(3»l) and (3 •2), the index "S" means that the coefficients ty and t,
are totaJ. coefficients of the lifting system, of a helicopter.
If the characteristics of the lifting system are calculated -with considera-
tion of Cx of the helicopter, then c^ in the first equation is assumed as equal
to zero.
Below, we will omit the index "E". For sinplicity, we will use the term
rotor instead of lifting system and helicopter instead of rotary wing aircraft.
The geometric rotor characteristics will be labeled by the solidity ratio o .
It follows from eq.(3.l) that, for a given value of V in helicopters with
identical rotors (equal a) and equal drag coefficients, the coefficient t^ has
an identical value. Since the aerodynamic rotor characteristics, i.e., the in-
terrelations of the quantities ty, t^ , m^ , V, and M^ are known (see Figs. 2. 142
to 2.145), it is possible to construct auxiliary graphs valid for all flight
conditions of a given helicopter and for all helicopters with_equal a, c^, and
Mq, by means of which - for any value of ty as a function of V" - we can find the
torque coefficient m^ , angle of attack of,, . f » and angle of rotor setting 9o
required for horizontal flight.
The sequence of constructing the auxiliary graphs for calculating horizontal
flight regimes will_be described for the Mi-4 helicopter with rectangular metal
blades (o = O.O63, c_, , = 0.009 with consideration of the rotor hub, or
' (a=o)
Cx. _ = 0.0075 without it). The calculation is made on the basis of experi-
mental aerodynamic characteristics of a rotor with rectangular metal blades,
a = 0.0525* When using these characteristics for the rotor of the Mi-4, the
conversion formulas are utilized (see Sect .6, Chapt.Il). In this case, the corir-
version is required because of differences in the rotors with respect to their
solidity ratio, and the difference in Mq must be allowed for. No differences
exist in profile or quality of blade manufacture, and both blade mass character-
istic and flapping conpensator are practically identical. The parasite drag of
the helicopter is taken without the rotor hub (the influence of the hub is taken
into account in the experimental characteristics of the rotor) .
Thus, conversion of the rotor characteristics is performed by the formulas:
(3.3)
(3.4)
(3.5)
(3.6)
325
^,=^x.
•
("1-')'^ .
4fi2K2 '
a:=aj
(0, — 0) ty
4B2F2
/Wf =
■■mt
i-^-^fico;
%-
=K
where t^ , ofi , m* , Qq are characteristics 'of the tested rotor.
The coefficients and angles without the subscript pertain to the Mi-4 /304
rotor.
The parasite drag coefficient of the Mi-4 helicopter entering eq.(3.l) was
determined from the curve of c^ = f (a^ ) obtained from full-scale wind-tunnel
tests of a helicopter without rotor (RLg.3.25)« If the angle of attack a at
zero deflection of the automatic pitch control is indicated on the aerodynamic
characteristics of the rotor, then the angle of attack of the fuselage is related
with the angle of attack of the rotor by the approximate expression (Fig. 3-26):
"/^a + V-A-",.-
(3.7)
Here, Sf is the angle of advance of the fuselage axis to the plane of rota-
tion, and DiKav is the difference in the angles of attack of the rotor at k ;^
and H = 0. For the Mi-4 helicopter, Sf = 5 and the quantity DiH^v is taken to
be e qual to -3° .
^x
-
""■T
1
^ ^
121
"-7
—
— -
Plane
V
of rotation
atX¥-U
f
N
>^-D,x.
i
d
s*^
■^ 'Automatic pi tch
f}^ control
^2Tr~\
1
/
-10
W ot)
Fusel age axis
Fig .3. 25 Parasite Drag Coefficient of Fig. 3. 26 For Determining the Fuselage
Mi-4 Helicopter vs. Fuselage Angle Angle of Attack.
of Attack.
Equations (3 '3) - (3'7) are used in the following sequence: For the se-
lected values of ty and V, prescribe the angle of attack of the rotor a and find
cxf , c'x, and t^ . Using eqs.(3«3) and (3 '4) i'or deteiTnining t^ and o-i from
the graph of the aerodynamic characteristics, check whether the values of t^
and QTi correspond. If not, assign a new value of a and again find t, and ofj .
Selection of the value of a can be done rapidly in practice. After determining
the final value of t^ , use the graphs of the aerodynamic characteristics to
find m^ and 0q , and determine b^ from eq.(6.18) of Chapter II by means of
the graphs in Figs. 2. 80 - 2.88. In this case, again make use of eq.(3.3) to
find the value of t^ corresponding to a = O.O9I, for which the graphs of Ztooo
are constructed:
326
t'=t.^
("o-^')'^
4B2K
i72
= t.
(0.091—0.063) 4
4-0.96V72
The calculation is carried out in Table 3»1«
In hovering flight, nit ^^.s determined also from the eDq^erimental curve
(Rig.2.1!fl) with conversion to a = O.O63 by the formulas:
tn.
rtti
(3.8)
V
a"
('
.-o)^J
4B2V2
^-t
r.3
(0, — 0) ty
4B2V/2
77l£,
8m,
TABUS 3.1
<y=0.14; 0=0.063; Afo=0.6
/3O5
^*-/
0.15
0,20
0.25
0.30
-3.5
—5.25
—8.0
—11.0
4.5
2.75
—3
0.007
0.0072
0.0077
0.0081
-0.0025
—0.00457
—0.00765
—0.01155
-0.00239
—0.00134
—0.00086
—0.0006
-0.00489
-0.00591
—0.00851
—0.01215
-0.98
—0.55
—0.35
—0.25
-4.5
-5.8
—8.35
—11.25
0.0055
0.00545
0.00645
0.00875
7.6
7.9
9.0
10.2
0.00008
0.0001
0.00015
0.0002
0.00558
0.00555
0.0066
0.00895
Having made similar calculations f Or a large range of ty , we construct
universal auxiliary graphs for determining the characteristics of the horizontal
flight of helicopters with a = O.O63, "c^, _ = 0.0075 (without a rotor hub),
and Mq = 0.6. These graphs are shown in Figs .3 '27 - 3 •29.
Some sinplification in the use of the auxiliary graph of m* , shown in
Fig. 3. 27, changing from plgrsical quantities to dimensionless and vice versa is
possible by constructing a graph in which the ordinate does not give m* ^ but
327
Z206
'"t^,^ 1 l ^z
""'" ^ ^ V .^
^^^ ^ K^*
:^ ^ s: V \% 7
\ v^^^ ^ n- z^
nnm ^ \ \ ^A -^ 4-L
aow ^ \^K^^7 ft
^^ V V C^^ ^J-t
\ \ nTV ^ Vl
\ \ -^^^^^ Till
mm<i~^ ^ V \<?, ^^-/ ■^J7'7
00075 ^ ^-"^ ^S" /2Z_
^ v>>>. t^^Tn
\ \ io\ , a/ /
^.-aX^^^"/ --</^
^^^ ^^ -
7-
^*-V
0W?5 1 1
0.75 0.20
O.W
0.20
Fig. 3. 27 AuxHiaiy Graph for Calcu-
lating Horizontal FUght Regimes:
Required_Power Coefficient as a Func-
tion of V, ty (Mo =.0.6; a = 0.063:
Fig. 3 .28 AiDciliary Graph for Calcu-
lating Horizontal Flight Regimes:
Setting Angle of Rotor (at k ='0.55;
Y = 4*85) as a Function of V and tj
=^(0=0)
= 0.0075 without rotor hub). (Mq = 0.6; a = 0.063; c,.^^ .^
without Rotor Hub).
= 0.0075
0.05 0.10 0.15 0.20 025 0.30
-5
-10
-h.f
s
'^
K3
1
1;
^
^
■>
\
;^
:>
'V__
s
^
^
\
^
ty=0.22
t
,=azX^
—
ty=ai;oj2;om
^ty =0.16; 0.18
Fig. 3. 29 Auxiliary Graph for Calculating Horizontal Flight
Regimes: Rotor Angle of Attack as a Function of V and t™
(Mo = 0.6; a = 0.063; c, = 0.0075 without Rotor Hub).
^(a=o)
328
the quantity +~ which is connected -with the physical quantities ty a rela-
tion having a sijoopler form than eq.(1.6):
i^t.
75Ar., e
U GaR
(3.9)
Such a graph is shown in Fig. 3 .30.
In Figs .3 '27 - 3 '30 the curves are plotted to values of V permissible for
the condition of flow separation at the rotor blades (see Figs. 2. 120 and 2.l2l).
The curves corresponding to ty = 0.24 were obtained by extrapolation of the ex-
perimental graphs.
At large Mq, when the con^Dressibility effect is appreciable and the auxili-
ary graphs become applicable only to the value of Mq for which they were con-
structed, it is expedient to plot, for heUccpters with a turboprop engine, a
graph for determining Mj,^, in reduced parameters: N^^ , = f(Vj. ) with the para- /307
r
meter G, for Mq = const (w^ = const). The reduced parameters are determined by
the formulas:
N,
h.fr
V p
V
Zo___ J So
7- 75 2
(0.379 -^)a/=-(2o,i yryy.
V
To
= const, nif.
%f'
Vr. = ^ 1 /-^= 20. 11/7^0 -^^const^M. =const2MoV^;
1/ / u h.f
(3.10)
Since, in the case of a turboprop engine, N^ determines the reduced fuel
/— — — , it is possible to construct auxiliary
consunption per hour G^
= G,
Po
graphs for determining G^^ and the relative fuel consunption per kilometer
r
q q
-p — = -jT- in the case of helLcqpters with turboprop engines.
AuxJli ary graph for the helicopter dynamic ceiling . From the minima of the
ciirves of the required torque coefficients (broken curve in Fig. 3. 27, designated
by Vh) we can construct a graph of (m^ . )Biin - f('fcy). This graph, shown in
Fig. 3. 31, can be used for determining the minimimi required power at any flight
altitude (at any ty) and for finding the theoretical dynamic ceiling of the heli-
copter Hjyn , i.e., the heights at which the available power is equal to the
minimum required power. The graph can also be used for determining the altitude
up to which horizontal flight is possible upon failure of some of the engines /308
of a multiengine helicopter.
329
0.075
0.050
0.025
-■
~"
--
7
"^
>,
\
^
s
\
—
UJ
^
^
^
N-
'I AC'
&
—
1
^
^
W
/
/
y//
>l
^
?>
.
>
(a
A/
0.76-^
V
^
^^«?<^
^/
Xo.16
ty=0.18
-^
^
^i?
^
^
^
—
4
>
h
O.IH
/■■
'
0.10
0.20
0.3 7
\f)min
-
0.010
-
_,
^
~
0.005
^
-^
^
'
\A.
A
0.10
0.15
0.20
Fig. 3 .30 Auxiliary Graph for Calculating
Horizontal FHght Regimes: Ratio — j^"''^ ■■■ as
a Function of V and ty (Mq = 0.6; a = 0.063;
c™, = 0.0075 without Rotor Hub).
*(a=o)
Fig. 3 .31 Coefficient of Minimum
Required Power of Helicopter as
a Function of ty (Mq = 0.6;
a = 0.063; Cx(q^o) = 0.0075
without Rotor Hub) .
Auxiliary graph for maxunum rate of climb . To calculate fUght regimes of
a helicopter in which the flight-path angle 9f i.p is not equal to zero, eqs.(1.6)
and (1.7; must be solved for 9fi.p after determining the value of mt with respect
to the available engine power for optimTaa rate of climb and after setting m^ =
for gliding in autorotation of the rotor. This problem is solved either with
the assunption of a small value of the angle 9fi.p (cos 9fi.p =1), or by suc-
cessive approximations; however, it is more convenient to construct a universal
aiEciliaiy graph.
First we determined the flying speed at which the vertical speed is maximum,
i.e., the optimum rate of climb Y^ . Calculations show that, for a helicopter,
the optimum rate of climb practically coincides with the rate of horizontal
fUght at which the required power is minimum. This is explained by the fact
that the excess of rotor shaft horsepower used for climbing is maximum in this
regime (since the available shaft horsepower of the rotor depends little on the
flying speed) and that the propulsive efficiency of the rotor (see Sect. 7,
Chapt.II), i.e., the efficiency of converting the excess rotor shaft horsepower
to an excess of propulsive power creating vertica.1 speed, depends very little on
the fljdng speed (with the exception of near-separation regimes). Therefore,
the optimum rate of climb for all values of ty is fo-und in Fig .3. 27 from the
curve connecting the minima of the required torque coefficients.
It is obvious that, for all values of ty, the regime of optimum climb cor-
responds to V = 0.15 - 0.2.
Therefore, the aioxiliary graph for determining the vertical speed of a
330
helicopter is constructed for two values of V: V = 0.15 and V = 0.2; for inter-
mediate values the vertical speed can be determined by interpolation. The
auxiliary graph is constructed in the following sequence:
Assign several values of 9fi.p (both positive and negative).
From eq.(l.6), find t^ , and determine Cx as a function of a^ :
9
«/=%^ -W;
Assign a mmiber of values of ty and find ty
(3.11)
(3.12)
From the graph of aerodynamic rotor characteristics with respect to t^
and ty , determine mt for all values of ty . ®
'9'
Then, determine Amt (see Fig. 3 .32):
Determine the vertical conponent of flying speed
Z202
(3.13)
v.=^=
l^sine^tp j-
V sin %iy.
(3.1^)
Construct the graph of Vy = f(Amt) with the parameter ty; such a graph
is shown in Fig. 3 .33.
It should be noted that, because of the linearity of the aerodynamic rotor
characteristics and because of the equidistant
translation of the ciirves of t^ i:qDon a variation
in a, the aixxiliaiy graph shown in Fig. 3 .33 is
applicable for calculating helicopters with any
c^x and CT (for Mg less than 0.6).
If the graph is constructed for large nega-
tive values of Am,., then the vertical rate of
descent of the helicopter during gliding in an
autorotation regime at a given peripheral rotor
speed can be determined.
To deteraiine the static ceiling of a heli-
Fig.3.32 For Determining copter and the rate of climb in vertical ascent,
the Increment in Power let us use the graph shown in Fig. 3 .34 which is
Coefficient in Flight a reconstructed graph of the aerodynamic rotor
along an Inclined Path. characteristics for V = 0.
tx
\
, 1
x
\
4m^
"
h'
X
\
mf
J
1
^
S. jr
<!
r^
2. Determination of Helicopter Performance Data
The sequence of determining the performance data of a helicopter from
331
/310
Fig. 3 •33 Atcd-liary Graph for Determining Maxim-urn
Rate of Climb.
^t / / /
^y / ^^
^: S^t/t^X.
,,.%jty/7.
'T-'^tK^^y
"•""' ^7^^,^
.^^^^#p i
^^^^^v!
„„„, .^^i^^^
0.005 y^>'^'^
gig--"
^^
0.05
a?o
0.f5
Pig«3»34_ Torque Coefficient as a Function of t
and Vy for V = (Mq = 0.6; a = O.O63).
332
auxiliary graphs Is as follows:
Select the design flight altitudes and calculate, for each altitude, the
lift coefficient in horizontal flight and the available power coefficient of the
rotor :
*'~ 1 ~'W^^W ^ "' A ' (3-15)
75m
-e^:.
'* QoF((o/?)3 (3.16)
The design flight altitudes are selected at intervals of 1000 - 1500 m.
The design altitudes should include the critical altitude and other salient
points of the altitude characteristics of the engine.
The torque coefficient m^ , angle of attack 0"^ . f , and angle of setting
9o , required for horizontal fUght of the helicopter are foimd for calculated
ty by interpolation from the auxiliary graphs in Figs .3 -27 - 3.30.
Maxunum and minimum flying speeds . These are determined from the inter-
section points of the curves oi'nit andm^ • There is no need to construct a
special graph of m^ , and m,. , and V ^ and Vnin can be found by direct /311
h • f d 1 s
interpolation from Figs .3 .27 - 3 •30. If the curves of m* and m,. at large
__ h • f d i s
V do not intersect (at the limit of separation m^ < mt ), then the maximum
b • f d i B
flying speed at this altitude is not limited by the available engine power but
by the separation of flow.
MaxLnum vertical, rate of cTi'Tnb . This is determined from the auxiliary
graph in Fig. 3 .33. Here, V^ and (mt , ) are found from Fig. 3. 27 for all
calculated flight altitudes, calculating
^'«'.ax = '"^a/^ -(/"...^Un. (3.17)
After determining Vy from the graph in Fig .3 .33, we find
Vy =P/o/?. (3.18)
max
As a typical exanple, let us determine maximxm and minimum speed, optimum
rate of climb, and maxLmimi vertical speed of the Mi-4 helicopter with an all-ip
weight of G = 7200 kg, tuR = I96 m/sec, and R = 10.5 m. All calculations are
given in Table 3.2, and the results are plotted in Fig .3 .35.
Practical and theoretical dynamic ceilings . These can be found from /312
333
TABLE 3.2
H,m
1000
1860
3500
5000
5500
h
0.138
0.152
0.166
0.195
0.229
0,243
^'"^««»V
1430
1500
1550
1315
1380
1300
£(('=0)
0.80
0.00836
0.00966
0.0109
0.84
0.01085
0.0134
0.0134
"'fe
0.008775
0.01015
0.0115
0,01145
0.0141
0.0141
Vn,ax
0.297
0.312
0.325
0.305
—
—
I'niax. km/hr
210
220
230
215
—
—
l^mln
0.035
0.033
0.03
0,09
0.103
0.122
l^mln, km/hf
25
23
21
63
72
85
v«
0.170
0.18
0.18
0.20
0.19
0.18
V.,. km/hr
120
127
127
141
134
127
'"h/mln
0.00537
0.00595
0.00665
0.00795
0.010
0.0109
im^
0.0034
0.0042
0.00485
0.0035
0.0041
0.0032
l^y
0.024
0.0277
0.0292
0.0179
0.0179
0.0132
''j'max- '"/s«i^
4.7
5.4
5.7
3.5
3.5
2.6
Fig. 3 .35: The former is the altitude at which Vy =0.5 m/sec and the latter,
the altitude at which V„ =0. From Fig .3 -35 we can determine, Xsj extrapola-
' m ax
tion, that the ceilings of the Mi-4 helicopter are equal to: H^y^ = 6400 m and
H,
dyn.
6550 m. These data can be found without constructing a graph of Vy
using instead the graph shown in Fig.3.31.
/•
'n^
,
—
-
\
-
-
'
-
r
/
5000
-
/
1*000
/
/
Vh
^
1
3000
/
/
/
^
V-
X
r
ZOOQ
'/n
n
'ma
-
-
mo
i
-
1
\
1
m
zoo y km/hr
Fig. 3 .35 Flight Characteristics of
Helicopter.
For this, the data in Table 3«2 are
used for plotting, in Fig.3.31,
the curves of m^
dli
tyVy
= m*
and m*
Mia
0.51
determining, by means of eq.(3.19), the relative air density
Tl "^diB y ouRn
against ty (the propulsive effi-
ciency of the rotor T| is deter-
mined from the graphs in Sect.?,
Chapt.Il). The values of ty at
which these curves intersect with
the curve of (mt j)Bin correspond
to the theoretical and practical
dynamic ceilings. Such construc-
tions are performed in Fig. 3. 36
from where we find that, at the
practical dynamic ceiling, we have
ty = 0.268 and, at the theoretical
dynamic ceiling, ty = 0.274» After
334
II
A =
(3.19)
from the standaixi atmosphere table or from the foimula
20-«JkmL
the ceilings are determined. In our exanple we have
which coincides with the values obtained above.
Static ceiling o f helicopter_ and rate of cj Limb in vertical ascent . These /313
are found from the auxLHary graph in
Fig. 3 .34, for which purpose the ciorve
^t
~~
"
-
-
-
^
/
. i-,t'^
—
i
7^
7^
—
/
/
-
^')>\
-
—
^
-
y
0.010
A'
s^v.^ -^
L f 0.5
^t tv — :; —
1^
..»>!
-^
■~
-
in
~
-
0.005
1
1
-
~
ri \\'
of mt
^is = f(ty) was plotted there.
0.70
0.15
0.20
0J5 t»
Fig. 3. 36 Determination of Practical
and Theoretical Dynamic Ceilings of
a Helicopter (Mq = 0.6; a = 0.063;
c". = 0.0075 without Rotor Hub).
(a=o)
The static ceiling of a heli-
copter is determined under maximum
engine operating conditions, since
transport helicopters are generally
not intended for prolonged hovering
and usToally hover briefly diuring take-
off and landing, closely above the
field in the zone of influence of the
air cushion.
As a typical exan^ile, let us de-
termine the static ceiling and vertical
rate of ascent of a helicopter at
takeoff power, with cuR = 212 m/sec .
The calculations, made ty means of the
graph shown in Fig. 3 .34, are given in
Table 3.3^
From the intersection of the
curve of m^ with the curve mt for ,
d 1 8
V = 0, we find ty corresponding to the static ceiling, and the static ceiling
itself: ty = 0.128; A
^'^^^^ = 0.917; H^t = 890 m.
0.128
For a more conplete study of helicopter data in hovering, a graph of maxi-
mum rotor thrust should be plotted as a function of flight altitude, for dif-
ferent tenperature conditions t^^t with and without consideration of the ground
effect (the latter is required for estimating the possibility of takeoff and
landing of a helicopter in mountainous terrain). The calculation (Table 3.3) is
335
H, m
h
0.1175
^'Vto' V
1700
'"'dis
0.00784
yy
0.001
(
V, mjsec
0.2
t
0.119
(
' taiXf k^
7287
TABLE 3.3
750
1000
1500
1860
0.126
0.13
0.136
0.141
1720
1685
0.8
1600
1560
0.00854
0.00858
0.00857
0.0086
0.002
—0,0005
—
—
0.4
-0.1
—
—
0.1285
0.129
0.129
0.130
7320
7170
6820
6630
performed by means of the 'graph in Fig .3 .34 in terms of the ciarve for Vy = 0:
Here, mt is determined from the available engine power, t is found from the
graph, and the maxim-um rotor thrust T^^x is then defined. The graph of Tj,ax for
the Mi-4 helicopter at takeoff power of the engine is shown in Pig. 3 .37. Con-
sideration of the groTind effect on the rotor thrust is acconplished by means of
the coefficient K^ which, for a given rotor, depends on the relative distance
to the ground h/R.
equal to
Thrust with consideration of the ground effect T^ , ^ is
(3.20)
In Fig .3 .37, Tg,g is determined during hovering of the helicopter at a dis-
tance of 2 m from the ground, when K^ « 1.12; this distance enables a helicopter
of the size of the Mi-4 to take off vertically and to change to forward flight
without touching, the ground (ground contact may take place during the takeoff
run when the pilot deflects the helicopter and it drops slightly).
The maximum range of horizontal flight \ t and maximum dioration of /314
horizontal flight Tjj . f are determined by the e^^jressions :
'Vi,
'■h.f,
±^
9inln
[km];
Ohr,
(3.21)
(3.22)
In these e^qsressions, we denote:
Gf = weight of the fuel consumed in horizontal flight of the heli-
^'^ copter;
Gjj = hourly fuel consunption in horizontal flight of a helicopter;
q = fuel consunption per kilometer in horizontal flight of the
helicopter:
^=-^- (3.23)
336
To determine the minim'um fuel
consunption per kilometer, the
minimum fuel consunption per hour,
and. the economic and cruising
speeds, we construct a graph of the
fuel consunption per hour and kilo-
meter as a function of flying
speed. To construct the graph, we
first use Figs. 3 .27 or 3.30 to find
the engine power required for hori-
zontal flight, and the engine char-
acteristics to find the fuel con-
sunption per hour.
The rotor rpm at cruising and
economic speeds should be estab-
lished beforehand. Usually these
are equal to the minimum permis-
sible ipm selected by the heli-
copter designer, on the basis of
flight safety and design considera-
tions; they should be combined with
the crviising regime of the engine.
For the Mi-4 helicopter, the peri-
pheral rotor speed in cruising and economic regimes is equal to aoR = 180 m/sec.
Calculation of the graph shown in Fig. 3 .38 is accomplished in Table 3.4 for an
average gross weight of G^v « G - ^ Gf = 69OO kg.
"^max l<S
8000 '
W=-30°
V
''4
-
—
V
sV
^
<-
'f
N.
V
%t
Z — 1900 L-n
7000
,W=3£
/"
^
S
''•<n
X
'*»^
r^^'i
6000
woo
mo Hm
Fig •3-37 Maximum Rotor Thrust of Heli-
copter in Hovering Flight.
TABIE 3.4
/3I5
V
V, km/hr
J, kffJm
Q
//=1000 m\ w/?=180 m/suc; Go,=6900 kj ; <y=0.172
0.10
65
0.0086
985
230
3.54
0.15
97
0.00725
830
175
1.8
0.20
0.225
0.25
0.30
130
146
162
194
0.00685
0.00715
0.0078
0.01015
785
819
893
1162
163
170
195
308
1.25
M64
1.203
1.587
It follows from the graph that the minimijm fuel cons-unption per hour and
kilometer and their corresponding cmjising and economic speeds are equal to
G,
^r„„„=163-^ at 14,^125^
337
The normal fuel load of the Mi-4 helicopter is 600 kg. From this amount,
we must subtract the fuel consumed for starting and ground testing of the engine,
for taxiing before takeoff, for test hovering, climbing, descending, and landing,
and also amount of fuel needed for
maneuvering in the air. The remainder
of the 'unconsumed fuel is incorporated
into the enpty weight of the helicopter
and is disregarded in defining the
fuel load.
In determining the above fuel cor>-
simption values, it is assumed that
engine testing takes 5 niin at low speed,
taxiing at an engine power of 0.3 of
the rated power takes 2 min (distance
0.3 - 0.5 km), test hovering and land-
ing at takeoff power takes 2-3 min,
climbing at the optimum rate takes
place at rated power, descent proceeds
at the most advantageous speed of Vy =
=4-5 m/sec at 0.3 - 0.5 of the rated
power. For transport helicopters, the
fuel needed for navigation is assimied
qfy/im
^hr ka/hr
\
\1
V
■-(
t
y
/
/
/
A
/
/
1.5
zoo
\
\
1.0
zoo
/
>
-
Cu,
y
^
■<L
r
y
Vs
'
0.5
wo
._.
50
100
150
Vkm/hr
Fig. 3. 38 Fuel Consimption per
Ho-ur and Kilometer of Helicopter.
as equal to 5% of the total fuel st^jply.
For the Mi-4 helicopter in long-distance flight at an altitude of 1000 m,
the sum of all fuel expenditin-es, together with the navigation svpply, amounts
to 100 - 115 kg, i.e., to about 15 - 20^ of the total fuel load. The path /3I6
and flying time consimied in climbing
and descending are, respectively, equal
to 20 km and 0.2 hr.
Mo' const,
(5.= const 2
Cx=constj
Tangent to
origin of
coordinate
Thus, the Mi-4 helicopter consumes
600 - 115 = I4S5 kg of fuel in horizontal
flight; the maximum range and endurance
of the helicopter are L^^f =
485 ,,^, 485
163
1.16
= 4I8 km, T,
h.f,
« 3 hr, while
endurance are ]
Tmax = 3.2 hr.
the technical range and
L„„ = 418 + 20 = 438 km.
Fig .3 .39 For Determining
and Economic Speeds
On the assunption that the specific
Cruising fuel consunption is independent of
engine power and that the power utiliza-
tion coefficient is independent of fly-
ing speed, the regimes corresponding to
maximum range and endurance can be detennined directly from the graphs in
Fig. 3. 27 in the manner shown in Fig. 3. 39.
Usually the optimum ipm in cruising and economic regimes is below that
338
selected by the helicopter designer. However, If the optlm-um rpm Is to be de-
termined, calculation of Nj,,f and Ghr Is performed for several values of cuR
h.f
and the optlmTjm rpm Is selected from this. On the assunption that the specific
fuel consunptlon is Independent of both engine power and engine rpm and if the
power utilization factor does not depend on the fljring speed, the maxlmtmi range
and duration can be determined from the following expressions:
270O, e 1 f ,
\ ^y /mln
''^A/niax~~ C.a^'^
The flying speeds and the corresponding rotor rpm (or V and ty) at which
iuX_ and » '' - reach a minimum can be found from the graphs of these quanti-
ties plotted on the basis of the graphs shown in Fig. 3 •27 •
nit
It should be noted that the quantity 1 / - ^z^ . ±q equal to the product of
"Dy V
helicopter performance and propulsive efficiency of the rotor K^T] [see eq.(7.lO)
in Chapt.IIj; consequently,
^V = ^^^^A^^lkm]. (3.240
Minimim ver tical rate of descent . This rate, in gliding in an autorota- /317
tion regime at a given peripheral rotor speed is deteraiined from the aioxLliary
graph shown in Fig. 3. 33. For this, eq.(3.2) is used for calculating t„ ; for
Amt = -(mt )nin and V^ , FLg.3-33 is used for determining Vy and then \ =
= VyOjR.
However, in autorotation the vertical rates of descent are determined in the
entire range of flying speeds, both at constant rotor rpm and at constant pitch
9o . To solve these problems, the graphs of rotor characteristics in autorota-
tion, shown in Fig. 2. HO, are used [if necessajry, these characteristics are
converted by eqs.C3»3) - (3 •6)].
For constant rotor ipm, the calculation is performed by the method of suc-
cessive approximations. As first approximation, we use cos 9fi.p = 0.97; after
calculating (ty \ by means of eq.(1.5), the quantities (t^ )i , a^. , o?, , c.
' 0*11
for a series of V are determined from the graphs of the rotor characteristics,
and the equations of motion of the helicopter are used for finding the angle
339
- F2
-^^6^.= ' , ° • (3.26)
After repeating the calculations tintil the values of the angle Qf i.^ coin-
cide, we find the flight-path speed and its vertical and horizontal conponents
V = V^R; (3.27)
Vy^VsinGfLp; (3.28)
K^=Kcos9^l.p. (3.29)
We note that -tan 6f i . p is equal to the inverse helicopter performance
during gliding in autorotation:
■ Un%fi^=-jr-. (3.30)
In calculating the autorotation regime with a selected rotor setting (usual-
ly 6o = 3-5°), the quantities ty , t^ , and 01^ are determined for several V
'O ' ' ■ ~ - ''o ' "o
from the rotor characteristics . Then, 0f 1 . p , U)R, and Mq are obtained from the
e35)ressions :
ta/iBflf=
72
0/?= A^^^^i^
a
If Mq is less than 0.6, the solution is considered valid since, in this
case, the effect of Mq on the rotor characteristics can be disregarded.
If Mn > 0.6, the calculations must be repeated, determining ty , tjt , and /318
C C
Qfg for Mq obtained in the preceding approximation. The successive approximations
are carried out rapidly and present no difficulties. After final determination
of u)R, we determine V, Vy , and V^ by means of eqs.(3.27) - (3.29).
As a typical exanple, let us calculate gliding in autorotation of the Mi-4
helicopter with a gross weight of 7200 kg at an altitude of H = for coR =
= 196 m/sec. The esqaerimental characteristics of the rotor converted to the
solidity ratio a « O.O63 are shown in Figs .3 .40 and 3.41.
The calciJ-ation is made in Table 3.5, and the dependence of Vy and Qti.-p on
340
dZ
ai5
0.1
A/
1/
\
i
'A
/
I
7
/
i)
/
-
^/
^
/
,/r 1
/
y
',
'
/
<\
4^
J
/
u
h
/
/
A
y
1
/
.^
/
/'
1
A
^
V
/
0.01
0.02
0.03
0.0't
\
1223.
Fig. 3 .40 Polars of Rotor in Autorotation Regime
(Mo = 0.6; a = O.O63).
"C?
\
\
s
\
s
N
s.
\.
■s^
-^
n
070
0.20
0.30 V
Fig. 3 .41 Angle of Attack of Rotor in Autorotation
Regime (Mq = 0.6; a = O.O63).
700
mVkm/hr
-10
V,m/s€e
-10
_
e/i.p
/
^
^
/
/
/
i^
\
-20
eh
\
Vy
\
■p
\,
\
Fig. 3 .42 Rate of Descent and Gliding Angle of Helicopter
in Autorotation Regime.
341
V is plotted in Pig. 3 .42. This diagram indicates that the minimum vertical rate
of descent of the helicopter is 7*2 m/sec at V = 130 km/hr, while the maximum
gliding range, equal to
Lgi-.
H
(3.31)
is obtained for 9f i ,
ml n
= -10°; (K^ ),,^ =5.7; V = 180 km/hr.
TABIE 3.5
//=0; o)/?=196m/5ec; /j,=0,138
V
0.15
0.20
0.25
0.30
y, km/hr
106
141
176
211
Cvc)'
0.1338
(<-r,)l
0.0315
0.0202
0.0166
0.0150
°c.
13
7
4.2
2.6
a/=«c.+8°
21
15
12.2
10.6
ci
0.0067
CxV-i
0.00239
0.00425
0.00664
0.00957
sin e^i.^
—0.253
—0.183
—0.174
—0.183
e/J./'
— 14°40'
— 10°35'
—10°
— 10°35'
cos ^jl,p
0.967
0.983
0.9845
0.983
h.
0,1334
0.1356
0.1358
0.1356
'-C
0.0315
0.0206
0.0168
0.0152
sin 8^/.^
—0.254
—0.183
—0.1726
—0.1826
^ftp
— 14°42'
— 10°35'
—10°
— 10°30'
Vy^V%\'a^flp,m/stc
-7.5
—7.2
-8.45
—10.7
3. Graphs for Determining Optimum Helicopter
Aerodynamic Parameters
Z22Q
The described method of aerodynamic design and the graphs of rotor charac-
teristics used in it are convenient for a check calculation of a helicopter v/ith
known parameters, since sufficient data are available for determining the coeffi-
342
cients ty and tj^ in calculating horizontal flight regimes and the coeffi-
cient mt. in calculating climbing regimes.
In designing a helicopter, a preliminaiy version of the parameters is se-
lected on the basis of practical experience vjith previous models and on the
"basis of applicable values of peripheral speed, thrust coefficient, load per
square meter of rotor disk, etc. The next step is to refine the helicopter
parameters. To study the effect of parameters on the performance data of a
helicopter, special graphs should be constructed. Such graphs are necessary
also in investigating the maximum possibilities of helicopters for inproving the
flight characteristics.
Calculations for aerodynamic parameter selection should be acconpanied by
weight calculations and by investigations of the variation of parameters in a
limited range within which the helicopter has a sufficient useful load.
In this Subsection, a graph is described to be used for defining the rotor
parameters ensuring the minimum required power (minimum fuel consiarption per
hour and kilometer) at given weight, ScxS, speed, and altitude. From this
graph, the optimum diameter, solidity ratio, and peripheral speed of the rotor
can be determined.
The equations for calculating horizontal flight regimes are trarieformed in
such a manner that, in all equations, the smallest number of sought parameters
will 'correlate dimensionless coefficients with the prescribed quantities . Equa-
tions (3.1) and (3.2) can be reduced to the form
'^CjS _ — <^aMg f^ 22)
In like manner, we transform the equation for determining the required power
N,^=±,i.Rf.Fm,^^ Q (aM,)3, ('%^) m,;
^n,t 1 3 '"fMoMJ/ (3.34)
2 c^s 150 —tj,
G ^ ot
It should be noted that the quantities Mq, Mf ^ , —, — — — are propor-
pa pa
tional to the reduced parameters of the helicopter: ou^ , V^ , Gp, N^^t .
It is obvious that the required power will be lowest at a minimimi of the
343
1=
TABIE 3.6
Mo
0.61
0.655
h
0.12
0.14
0.16
0.18
0.12
0,14
0.16
0,18
tx
-0.00853
-0.00995
-0.01138
-0,0128
—0.00853
-0,00995
-0.01138
-0.0128
Wf
0.00825
0.C0957
0.01157
0.0146
0.0086
0,0098
0.01165
0.0143
8.
o
—tx
0,03056
0.03039
0.03214
0.0360
0.0342
0.0334
0.0347
0.0379
II
0.00558
0.00651
0.00744
0.00837
0.00643
0.0075
0.00857
0.00965
Mjc
0.00024
0.00033
0.00043
0.00055
0.00028
0.00038
0.00050
0.00063
tx (»=0.091)
-0.00877
—0.01028
—0.01181
-0.01335
-0.00881
—0,01033
-0.01188
—0.01343
Mf
0.0084
0.00972
0.01178
0.01483
0.00872
0,00993
0.01188
0.0146
O
-tx
0.0312
0.03085
0.0327
0.0366
0.03465
0.0338
i
0.0354
0.0387
-tjcoMl
0.00614
0.00715
0.00817
0.0092
0.00706
0.00824
0.00942
0.0106
Mji
!
ratio
mtMoMfi
To find it, a graph in coordinates
■n_ = f I
t^CTM^
IS
t M^ " " '■"'"
plotted for a value of the ratio — Z¥~~ S^^^^ ^y eq'(3«33) at Mf^ = V/a
= const. '^
Z222
The sequence of constructing the graph is as follows: For the value of Mf i
selected for the investigation, define the aerodynamic characteristics of the
rotor in the form of a dependence, shown in Figs. 2. 105 - 2.109, for several
values of Mq . After assigning several values to the coefficient ty and the
G
-, determine t^ from eq.(3.33) and find mt for each M©
from the graphs of the aerodynamic characteristics. Then, calculate the ratio
of the coefficients entering eqs.(3«32) and (3*34). When using the solidity
ratio of the rotor, the quantity t, is converted by eq.(3.3) or by the formula
tx, = tx,-
(°i — °2) <yMo
4fi2M5l
In Table 3.6 a calculation is made for a flying speed of V = 275 km/hr at
n
an altitude of H = 1000 m (a = 336.1 m/sec, i pa^ = 6400 kg/m^ ) for -- — — =
= 4670 kg/m^. For these data, we have ^^
-L.=t.
Mfi =0.227;
0,2272.6400
4670
0.071 1/„
TABLE 3.7
Assigned Parameter
0=0.091; <^R=2\2 m Isic ;
Mo=0.63 (point a)
Vc S
0=34.56m; -H^L. = 0.008;
F
0=0.091 ( point b)
D=34.56 m ;
. Sc^S_
= 0.008;
ioR=220 m /sic; Mo=0.655
( point c)
Optimum
Parameter
/ 150AV,/_\
hp
(1>'^^\ _o.00689;
0.0317
6816
\ f /opt
D„^^=£37.43 m
Mo^,y.=0.63; <^R,^t =
'^=212aw/5«c-
0,0331
7117
oo^ =0.095
0.0336
7224
The graph for determining the optimum aerodynamic parameters is shown in
Fig. 3 .43. Curve 1 connects the minima of the curves with identical o. From
345
curve 1 we find the optimuiii rotor diameter at a given rpm (Mq) and a. Curve 2
is the envelope of the cijrves with identical ct, from which we find the optimum
rpm at given diameter and a» Curve 3 is the envelope of the curves with identi-
cal Mq, from which we find the optimian solidity ratio at given diameter and Mq .
As an exanple. Table 3*7 gives the optimum parameters of a helicopter for
G = 35,000 kg and Sc^S = 7-5 m^.
"l ,
\
\
\,
s
-
—
^
f
rc^Sf^pa
-j(;
4-
\°^^:
N
>
nnuf)
\
K-
\
P
y^
.J
/,
i-a
o,ato
s
^
S
L
/
r^
~
~-
—
\
V
-
^
\
•^
—
\v\
/_
V
/
\
\
\^^
i
^T'
A
'J
V
—
-
—
UJJjj
KA
N
*^
N
^
X
"^\
K
k
^
/;
//
n
c
• ty= 0.1
0.12
It o.n
-A 0.16
xty-0.18
"Mil
\
s
V
r.
y
/
b
^^
-/
^
0.030
1
~
-
—
—
-
-
—
^
6 = 0.10
0.0Z5
J
7323
0.0025
0.0050
0.0075
0.010 ECxS
Fig. 3 .43 Graph for Determining the Optimum Aerodynamic Para-
G
meters of a Helicopter (Mfi = 0.227;
p p a. 2-1 Cj^ iD
= 0.73).
The above method can be used for finding the optimum aerodynamic parameters
of a helicopter with a tractor propeller and wing; however, in this case, it is
necessary to first determine the parameters of the prcpeller and wing (m^ ,
t„ /t„ , etc.) at which the dependence tv„ = f(mtv.) is optimum, i.e., at which
the smallest values of t^y for all mt„ and ty„ are obtained.
Section 4« Aerodynamic Design of a HeUcopter Based on
Concepts of Rotor Performance and Efficiency
The concepts of perfonnance K and propulsive efficiency T| of a rotor are
given in Chapter II, Section 7« There graphs are presented, obtained from ex-
periment and calculation and useful for finding the values of K and Tj.
In this Section, we present a method of aerodynamic design of a helicopter
346
with the use of the concepts of performaxice K axid efficiency 71. The design
formulas for determining required power and vertical speed of a helicopter are
conpletely analogous to the formulas for calculating airplanes.
This is a very sinple method of calculation, easily extended to helicopters
of any configuration with a wing and tractor propeller or cruise jet engine.
In a general form, it pennits making various estimate calculations in a sinple
manner: estimating the e^^jediency of installing a wing and tractor propellers
on a helicopter, finding the power/weight ratio M/G required for producing /32?^
a given maxinum speed, and determining then the amount t)y which to reduce the
required power when reducing the parasite drag of the helicopter.
Since the performance and efficiency yield an approximate description of
the aerodynamic characteristics of the rotor, this method of calculation ranges
high among the approximate methods of aerodynamic design.
1. Helicopter Performance
The helicopter performance in horizontal flight regime is determined ty
eq.(7.9) of Chapter II
ty ty Cy
^>'^ t^-t^ ^ r^"" cx + 'cj^' ' (4.1)
In calculations it is more convenient to use the inverse quantity, namely
the inverse performance of the helicopter:
i=t+V=^+V- (4-2)
Changing to dimensional quantities, Y = G and Qp^r > we obtain
--=— +^', (4.3)
where the parasite drag of the helicopter is
When using the conversion formulas for determining the performance of a
rotor with differing parameters (see Chapt.II, Sect. 7. 6), the helicopter per-
formance is found from the expression
1 1 , (» — 'O^y AmprTi Qo„.
A:^ K 4B2k2
347
2. Performance of Multirotor _ and CoinDOsite _Helicopters
In the general case, the inverse performance of the craft is
1 ItX+Q^, ^X Q^
K,
^y
a '
(4.6)
where SY and Sx are the sums of lifts and drag of all lifting elements of the
helicopter.
Let us derive the expressions of EY and EX, for two types of helicopters -
Single- rotor helicopter with wiuR . The lift of the helicopter lifting
system consists of the sum of lifts of the rotor and wing
J,v==Vr,,+y^
(4.7)
¥e represent Ey in the formZ/Y= G(Y^ot + Y^ ) having designated: Y^ot =
^'°^ and Y„ - ^"
Sy " Ey
In horizontal flight, we have
'ret 'ryv ^ 1 ;
V , ^rot ^rot .
rot — „ : — .
(4.8)
y^ =■
'y^ ^y^ 5^ —2 I
(4.9)
h^ F"
V'
The drag is made up of the drags of the isolated rotor and wing and of pro-
jections of the rotor and wing lifts onto the direction of motion (see Fig. 3. 11)
^X = X,,,^X^-^Y^,t,a^, + y„ Aa^, (4.10)
where Affro t ^-^-^ ^'^w ^^® "^^^ averaged downwash angles of the rotor and wing.
On substituting eqs.(4.8) and (4.10) into eq.(4.6), we obtain
Qpar
^=^..(i+AM + ^.(^+Aa.)+^'
(4.11)
348
The downwash angles are determined, as described in Section 1, by the ex-
pressions
^^ivt=*ret
jiX
t¥
Aa^=0.26x^^.
(4.12)
Two- rotor helicopter vri.th wing » After performing similar calculations, we
find
where the subscripts "1" and "2" denote quantities pertaining to each of the
rotors .
The total downwash angles due to the other two elements of the lifting
system of the helicopter are equal to
Aa.,r.=x„^. ^+0.26x1^',
l^^nt, = "roi. -7- + 0.26x2 ^ ,
t,a^ =0.26x^. ^+0.26x^. "^
(4.1!).)
For a helicopter of side-by-side configuration, both rotors operate under
equal conditions (all quantities with the subscripts "1" and "2" are equal to
each other), and k^ = Kg = Hg , g . Therefore, for a helicopter of side-by-side
configuration we obtain the following expressions:
Qpar
a '
^^rvt — ^ivt
aty
nK,
- + 0.26x^-1^;
a/y
Aa^=2.0.26x, ^.
(4.15)
/326
(4.16)
(4.17)
For a helicopter of fore-and-aft configuration with a wing between the
rotors, t?ie fixjnt rotor is virtually outside the influence of the tail rotor and
wing, and the wing is outside the influence of the tail rotor. However, the
induction coefficients h for the tail rotor and for the wing should be doubled
349
(4.18)
The total required power of rotors of a fore-and-aft helicopter depends
Uttle on the relation of rotor lifts. This is e^^jlained by the fact that, in
conformity with the general theory of induced drag, this power does not depend
on the lift distribution between individual elements of the lifting system, and
the profile power of the rotors does not greatly depend on the lifting force
of the rotors (in regimes not close to flow separation). Therefore, to detei^
mine the total required power of two rotors we can set Y^ = Yg . Actually, the
lifts of both rotors are close in value with respect to balancing conditions of
the helicopter.
After setting %otj_ = %ot^ = Y^ot and K^ = K3 = K in eqs.(4.13) and (4.14),
we find that the quantity \ can be determined by eqs.(4.l6) and (4.17), with
the induction coefficients not doubled. The physical meaning of this expression
is that, to determine the total power, it is possible to replace two rotors by
one with a double Hfting force inserted between the rotors. The downwash of
this rotor is equal to the half-sum of the downwashes of the front and tail
rotors, i.e., equal to half of the downwash of the tail rotor.
The sequence of calculation of helicopter performance is as follows: In a
check calculation of a helicopter the gross weight, diameter, solidity ratio,
rotor rpm, and parasite drag coefficient are known. After assigning the flying
speed and altitude, find the following dimensionless coefficients:
iyi
— Q0/=-(a)/?)2
M — "^
a
on the basis of which, using the graphs in Section 7, Chapter II, find the rotor
performance. Then, calculate Qp^r from eq.(4.5) and determine K^ .
In calculating the performance of conposite helicopters, it is necessary /327
to know the lift distribution between individual elements of the lifting system,
i.e., Yrot , ^rot > ^w • ^o^ estimate calculations, we can assign Y„ and Cy
for some flight regime, bearing in mind that these quantities can be obtained by
an appropriate selection of the setting angle and the wing area. Then, using
eqs.(4.8) and (4.9), we find Y *, ty • from the rotor and wing characteristics
rot
we determine K, K„. After calculating the downwash angles by eqs.(4.17), we
350
find Kj, .
When the geometric characteristics and the setting angle of the wing are
given, the following method can be used for determining rotor and wing lift in
horizontal flight.
The angle of pitch of a two-rotor helicopter, measured from the plane of
rotation of the rotor (front rotor for a helicopter of fore-and-aft configura-
tion), is determined by the following e^^jression:
2y„f a 2Y^t
Equation (4.19) is obtaified from the condition of equating to zero the sum
of projections of all forces onto the direction of motion, on the assunption _
that the angle ^ = a^ = Ci is small, Ti = Tg, Hi = Hg = H, + TDik, H, « 0.35 VT,
Yrot = T, X,„, = Tt? + H:Ti^ + T3 (,? + e,„J + % + H3 + Qp„ + X„ » EX = 0.
From the angle of pitch of the helicopter, we can find the angle of attack
of the wing
aK'=^ + 2«r -Aa^, (4.20)
where s„ is the setting angle of the wing relative to the plane of rotation of
the rotor.
— S — Q
For known V, ty„, -r^t t„ , — Fr^^» e„, e^ot » ^ih (the desired value of
Zi ro Li '
D-yH is obtained by selecting the angle of stabilizer setting), using eqs.(4.19)
and (4.20), as well as (4.9)» (4.14;, and (4.15), all quantities enteririg these
formulas are fo\ind by successive approximations: a„ , «?, °y„ » ^y ^j ^w* ^rot*
We re_commend the following sequence of calculation: After assigning q?„ ,
find c„ , Y„ , K„; by means of eq.(4.15) determine 2i.ro\.> and then ty aq, .
find «? and, from eq.(4.20), determine a^ of the second approximation.
Two or three approximations must be performed. In this manner all quanti-
ties for calculating K^ can be obtained.
As an exatiple, Table 3.8 gives a calculation of the reciprocal performance
of the Mi-4 helicopter. The initial data of this helicopter are given in Sec-
tion 3. Performance and efficiency of the rotor were determined from the graphs
in Figs. 2. 159 and 2.160, with conversion to the difference in solidity ratio.
The difference in blade profiles for Mg = 0.6 can be disregarded.
The results of calctilating -=z — for the entire range of t„ are plotted in
Fig. 3 .4^, indicating that the inverse performance of the helicopter is minimal
at V = 0.25 - 0.3 and at a lift coefficient close to the maximum permissible
owing to flow separation.
351
The maximum performance is K^ =6.0. At small V, the reciprocal per-
formance of a helicopter increases owing to a decrease in rotor performance and
at large V, owing to an increase in helicopter drag.
TABLE 3.$
(0.063— 0.091) ^y
/228
K^ Ka^ 4-0.96K2
Y
0.15
0.20
X'.
3,5
5.4
T
1.011
0.980
VK,,
0.286
0.1854
0.063—0.091
4-0,96K2 '"
—0.0462
-0.0261
VK
0.2398
0.1593
Cjc
0.009
0.009
0.063<y
0.023
0.0408
MK,
0.2628
0.2001
^ 0.063 <y
0.25
6.95
0.972
0.144
—0.0167
0.1273
0.009
0.0638
0.1911
^ = 0.14
0.30
0.35
8.5
9.25
0.964
0.947
0.1176
0.108
-0.0116
—0.0085
0.106
0.0995
0.0095
0.01
0.0980
0.139
0.204
0.2385
A second exanple of calctilation is that of the performance of helicopters
of different conf igtirations : single-rotor, fore-and-aft, tandem, single- rotor
with wing, and tandem with wing.
The calculations were made under the following conditions: For helicopters
without a wing, the lift coefficient of the rotor is equal to ty = 0.13, and
rot
for helicopters with a wing to ty^ = O.I6 and 0.32. The larger value of ty
for helicopters with a wing corresponds to two cases: a decrease in rotor dia-
meter when a wing is installed and a decrease in peripheral speed without a
change in rotor diameter. The soUdity ratio of the rotor is a = O.O9I, Mq =
= 0.65, and Mo = 0.65 - / ■ ^'j'l - = 0.587 in the latter case. The angle of wing
setting was selected so that relief of the rotor load was equal to at least 20^
at M,i > 0.2. The performance and efficiency of the rotor were determined 7329
from the graphs in Figs. 2. 159 and 2.160 and both Cy and wing performance, from
Fig. 3 .45* The parasite drag coefficient of the single- rotor helicopter, re-
ferred to rotor area, is equal to 0.0075 and, on a decrease in diameter, becomes
0.0075 — ^ — = 0.00925; for two-rotor helicopters, the magnitude of Ec^S is
twice that of the single- rotor helicopter. The wing area of the single- rotor
helicopter, referred to rotor area, is equal to 0.0325; on a decrease in rotor
diameter, the wing area did not change and in relative values was equal to
352
0.0325 -°lM_ = 0.04.
u •Jo
The relative wing span t^/R is equal to 0-85 and 0.95,
respectively. For tandem helicopters, the wing area is determined by the rotor
dimensions and is assrimed as O.I6 of the area of one rotor. The aspect ratio
of the wings is equal to \„ = 7.2.
0.Z5
0.20
0.15
Im
K„
..
/
^
1 n
20
/
"■\
/
V
f
\,
/
V
>.
,/
\
"/
\
n c
10
/
\
/
\
/
\
/
A
/
/
10
zn oc„
Fig.3.Z|4 Reciprocal Performance of Fig .3 .45 lift Coefficient and Wing
Helicopter as a Fimction of lift Coef- Performance as a Function of Angle
ficient and Relative Flying Speed. of Attack.
Calciilation of the performance of helicopters without a wing is made in
Table 3.9, while the performance of helicopters with a wing in a version with a
decreased rotor diameter is given in Table 3.10.
JL
0.3
0.2
0.1
T
n
1
-
\
\
\
vi
7"
\
,1
-
1 '
\}
\,
H
y
—
\
-A
v5
^
v;
i
V.
—
4
^
i^
%
^
■^z
—
':
■-"
^
1
0.1
0.1 Mfi
Legend :
Single-rotor configuration;
Side-by-side config-uration;
Fore-and-aft configuration;
wo/w, helicopter without wing;
I, helicopter with wing and
reduced diameter;
II, helicopter with wing and
reduced peripheral speed.
Fig. 3 .46 Reciprocal Performance of
Helicopters of Different Configura-
tions with and without a Wing as
a Function of Mji .
353
I I
TABLE 3.9
HELICOPTERS WITHDUT WING
V
0.15
0.20
K
3.5
5.2
1
1.00
0.977
'CxV^
0.0142
0.0253
''yE
Single- rotor configuration
1
0.286
0.1925
l^rot
1
0.300
0.2178
Xh
Fore-and-aft configuration
^rot
0.143
0.0962
Xco = 0.65
^,^ = 0.26
K
ant.
0.181
0.102
)'™^J^ +AawfJ
0.2335
0.1472
1
0.3907
0.2687
_ '<h
—0.0558
—0.0314 -
Side- by- side configuration
^p.rot
x„ = -0.4
'yj,=0.26
+ AaTOf
0.2302
0.1611
1
0.2444
0.1864
0.30
7.85
0.962
0.057
0.127
0.184
0.0635
0.0454
0.0862
0.2092
-0.01395
0.11305
0.17005
0.40
8.42
0.936
0.1014
0.119
0.220
0.0595
0.0255
0,0722
0.2373
-0.00785
0.11075
0.21215
The results of the calculations are plotted in RLg.3.46, which shows that,
in the entire speed range, the reciprocal performance of the helicopter of side-
by-side configuration has a lower value and that of the fore-and-aft configura-
tion, a higher value. _The maximtim performance is equal to: 6 for a side- /33I
by-side helicopter at V = 0.27; 5*5 for a single- rotor helicopter at V = 0.29;
4.8 for a fore-and-aft helicopter at V =0.3. At M,i = 0.26 (V = 0.4), the per-
formance of the helicopters is, respectively, equal to: 4*7; 4*55; and 4'23..
The wing, relieving 20 - 30% of the rotor load at high flying speeds,
changes the helicopter performance in the following manner: If, on installation
of a wing, the rotor diameter was decreased, the helicopter performance increases
very little (ciorve I). If the rotor diameter was not decreased but its ipm was
raised (curve II), the maximTjn performance of the helicopter increases by
0.5 - 0.9 (by 10 - 13%) and, at maximum speed (Mf^ = 0.26), increases by 0.4
( approximately ^%) . Calculations showed that if, on installing a wing, the
rotor parameters are not changed so that the rotor at high speeds has a very low
354
TABLE 3.10
'^s
Single-Rotor Helicopter with ^ing
= 0.16; c^ = 0.00925; iMo = 0.65; 5»v = 0,04; 7*v = 0.95; x,,f = 0.2;
%H, =0.69; £„, =21.97°
Tandem Helicopter with Wing
<ys = 0.32; c; = 0.0185; Mo=0.65;"S^=07l6;r=7^ =
= 1.9; «£4=— 0.4; »rrf = 0.12; x^ =0.4; i^ = 16.3°
AoC|y, radiui
A a^t- (from »ing)
Act/V^ (from interference)
Krot
1
+ Aa,
0.15
—0.45
12.36
0.795
19
0.112
0.049
0.951
0.152
3.45
0.0065
0.0065
0.2963
0.1644
0.282
0.00809
0.0143
0.20
-2.48
14.63
0.915
15.8
0.0595
0.10
0.90
0.144
5.3
0.0075
0.0075
0.1961
0.1228
0.176
O.0123
0.0254
0.214
0.30
0.40
-7.10
-12.66
12.7
8.23
0.810
0.545
18.4
23.8
0.0235
0.0126
0.200
0.24
0.80
0.76
0.128
0.122
7.8
8.05
0.0066
0.0045
0,0066
0.1348
0.0778
0.1078
0.0156
0.0571
0.180
0.0045
0.1287
0.0546
0.0979
0,031
0.1015
0.212
0.15
-0.6
8.4
0.548
22.5
0.1276
0.068
0.932
0.149
3.5
0.00145
-0.0640
-0.06255
0.2274
0.172
0.212
0.0117
0.0143
0.2.38
0.20
-2.9
9.7
0.63
21.8
0.066
0.139
0.861
0.1376
5.23
0.00167
—0.0332
-0.03153
0.1595
0,1119
0.1373
0.0156
0.0254
0.1813
0.30
—7.62
7,22
0.48
22.8
0.0262
0.237
0.763
0.122
7.6
0.00126
-0.0131
—0.01184
0.1198
0.0701
0.0915
0.0166
0.0571
0.1652
0.40
-12.5
2.95
0.22
15.9
0.0155
0.195
0.805
0.1286
8.3
0.00058
-0.00777
-0.00719
0.1134
• 0.0783
0.0913
0.0153
0.1015
0.208!
thrust coefficient, then installation of a wing will not result in a decrease in
required power.
It should be noted that an increase in thrust coefficient ty„ for a heli-
copter with a wing leads to a_decrease in its dynamic ceiling. This is so since,
at the optimum rate of climb Vo ~ 0.2, the wing only insignificantly relieves
the rotor load, and ty acquires the maximum
permissible (in view of flow separation) value
at a lower altitude. Furthermore, at large
0.2
0.1
r~
^
<'
■^.
/
f
\
/
I
/
1
/'
•
>^
1
/
4
/
\
\
/
f
/
/
^
/i
/
/^
/
/
/
V
'/
//
a
I
!/,
?
//
('■
flow separation at the rotor may occur at
is still small. To
low flying speeds when ty
reduce t„ at these altitudes, a 5 - 8/
'rot » ' /
crease in rotor rpm can be advantageous.
m-
It follows from Table 3.10 that the wing
performance with consideration of downwash by
1
the rotor
(^
K,
+ ba„j decreases by
0.1
0.1
^fi
Fig. 3. 47 Relative Lift of
Wing for Helicopters of
Different Conf igurat ions .
Legend :
Single-rotor con-
figuration;
Side-by-side con-
figuration;
Helicopter with wing
and decreased dia-
meter;
II - Helicopter with wing
and decreased peri-
pheral speed.
I -
several units at high flying speed and even
more at low speed. The wing, producing down-
wash near the rotor, somewhat reduces its per-
formance. This explains the slight change in
helicopter performance when a wing is installed.
On a helicopter without a tractor pro- /332
peller, a wing without a f jjxed angle of setting
has a maximum angle of attack «» in horizontal
flight at V = 0.2 - 0.15. At smaller V, this
angle decreases owing to an increase in down-
wash from the rotor; at larger values, it de-
creases due to an increase in pitch angle of
the helicopter. Therefore, when a wing has a
small area and large angles of attack, its lift
increases at high speed despite a decrease in
a^ , but insignificantly (Fig. 3. 47, single-
rotor configuration). Conversely, if the wing
has a large area and small a„ ( side-by-side
configuration), then at large speeds Cy
markedly decreases and the lift becomes less than at average speeds.
Thus, on helicopters without a tractor propeller or other propeller, the
wing should have a small area and large a„ or be provided with mechanization for
controlling the amount of Cy .
In a climbing regime, the angle of attack of the wing decreases, while it
increases in gliding. At a fixed angle of wing setting in an autorotation
regime, flow separation from the wing is inevitable, which can be tolerated in
the presence of a small wing lift ( small wing area and reduction in Cy by
356
mechanization of the vrlng) .
3. Det emanation ofH ellcopter Flight Data
If both helicopter performance and rotor efficiency are known, the required
power of a helicopter is determined by the expression (see Sect. 7, CHiapt.Il):
^V
OVlmlsec] 1 (4.21)
755 Ki,r\
°^ ^ _ aVlkmlhr] J_
2705 /C.r,- (^^22)
The sequence of calculation for helicopters of various configurations is
described in Subsection 2. The rotor efficiency is determined from the graphs
given in Section 7> Chapter II. Consequently, on assigning the flying speed /333
and altitude, a graph of the required power of the helicopter can be plotted.
In hovering flight, the required power is determined from aerodynamic charac-
teristics of the rotor in a hovering regime: Nh.f is calculated at all flight
altitudes under the condition T = G.
The maximum and minimum flying speeds are determined from the points of
intersection of the cvirves of required and disposable power. At all flight
altitudes we must find the maximum permissible speed Vpg, with respect to flow
separation conditions; if V„ ax > ^part then the flying speed of the helicopter
is limited by the value of Vpe , .
Having plotted the cvirves of required power and knowing the engine charac-
teristics with respect to fuel consumption, the fuel consumption of the heli-
copter per hour and kilometer can be plotted as a function of flying speed (see
Fig. 3. 38) and, as described in Section 3, the maximum range and endurance,
cruising and economic flying speeds can be determined.
If the helicopter flight path is inclined, the propulsive force of the
rotor should balance the projection of helicopter weight onto the direction of
flight, which is equal to G sin Qfi.p or G ^= — (see Pig. 3.1). Therefore, the
expression for engine power takes the form
^=^^(T+^+~J-^*-^ + 755x1 • (4.23)
It follows from eq.(4»23) that the maximum rate of climb of a helicopter is
determined by the formula
"max Q
357
^.fV
2000
1500
WOO
ZOO V kmjhr
Fig. 3.4s Required and Available Horsepower
of Helicopter.
ams
0.0005
0M025
Fig. 3. 49 Ratio Nh.f/Ga for Helicopters of
Different Configurations with and without
Wing, as a Function of Mfj .
Legend:
Single-rotor configuration;
Side-by-side configuration;
Fore-and-aft configuration;
wo/w - Helicopter without wing;
I - Helicopter with wing and decreased
diameter;
II - Helicopter with wing and decreased
peripheral speed.
The optimum rate of climb and minimum power consumption ^.f^^ are found
from the graph of required powers. After determining Vy at all altitudes and
constructing the graph Vy = f(H), the dynamic celling of the helicopter is
determined by graphical means (see Fig. 3. 35).
From eq.(4.24) for \is = 0, the minimum rate of descent of a helicopter
in an autorotation regime of the rotor (Vy )Bin is derived.
To determine the angles of attack a and angles of setting 9o of the rotor
it is necessary to calculate the coefficient of propulsive force of the rotor
Qpar+G~
^'~ r (4.25)
and, knowing ty and t^ , the angles o* and Go must be determined from the graphs
(see Figs. 2.63 - 2.70 and 2.105 - 2.109) or from eq.(3.95) given in Section 3,
Chapter II.
As an example, let us carry out an aerodynamic calculation of the Ml-4
helicopter with rectangular metal blades. The graph of helicopter performance
is given in Fig. 3. 44- The graph of the required and disposable powers is shown
in Fig. 3. 48 for six flight altitudes.
At V = 0, Nh.f is determined by the expression:
where mt is found from the graphs shown in Fig. 3. 34 for Vy = 0.
The maximum vertical rates of climb and minimum rates of descent in an /335
autorotation regime are calculated in Table 3.11.
Determination of the other flight data is accomplished by means of the
graph in Fig.3.Z|B.
A comparison of N^.f calculated by the auxiliary graph in Fig. 3. 2? with
Mh. f found from helicopter performance and efficiency shows satisfactory agree-
ment: Vv is also close in magnitude.
''max
Figure 3.49 shows a graph of required power based on Ga for helicopters of
various configurations. The graph is calculated by means of the helicopter
Nh t
performance graph given in Fig. 3. 46. The ratio — -^ — is determined by the
formulas :
Qa 75iKi,r] K'^-'^U
359
in forward flight, and by
in hovering flight,
Ga
1
755
/njMo
TABIE 3.11
(4.28)
H
V„, km/hf
V«
h
fj, .
^^ycL ,„. mlsic
1430
115
0.163
0.138
1.010
880
550
4.86
—7.8
1000
1500
115
0.163
0.152
1.002
865
635
5.57
—7.6
1860
1550
120
0.170
0.165
0.989
8ti7
693
6.0
—7.4
3500
5000
1315
1380
125
125
0.177
0.177
0.195
0.229
0.953
0.850
865
970
450
410
3.75
3.05
-7.2
—7.2
5500
1300
115
0.163
0.242
0.850
1060
240
1.83
—7.9
The power utilization factor § was taken as equal to O.93_for two-rotor
helicopters, and as 5 = 0.88 for a single-rotor helicopter at V s 0.15 and as
§ = 0.83 at V = 0. In hovering flight, the helicopter wing is swept by the
rotor and creates a negative lift; therefore, at V = the lift coefficient of
the rotor increases by 2$ for the single-rotor helicopter with a wing and by B%
for the helicopter of side-by-side configuration with a wing; the value found
from eq.(4.28) increases accordingly.
We see from the graph that, because of a difference in 5 in hovering
regime, the ratio N/G is lower for two-rotor helicopters than for single-rotor
versions.
The largest value of N/G refers to helicopters with a wing and with a re-
duced rotor diameter, while the smallest value refers to helicopters with a wing
and with reduced rpm for the single-rotor helicopter and for the side-by-side
helicopter without a wing.
Thus, to ensure the possibility of hovering, the helicopters in question
should have a different engine power per kilogram of gross weight. Correspond-
'ingly, they will have different flight data in forward flight. Table 3.12 gives
some flight characteristics of helicopters which were obtained in our example /336
at a flying altitude H = for the following characteristics of the engine:
Nt.
N,
ho V »
N„
= 0.85 Nt
N=
= 0.7 Nt.
The maximum vertical speed of the helicopter was determined by the formula
360
V
"dis
^h,,
y...= '^^^'^{-Oa Oa
(4.29)
The minimum rate of descent in an autorotation regime (Vy„),i„ was deter
mined for \i, =0.
TABLE 3.12
Helicopter
Configuration
Side- by- aide
Single- roior
Fore- «nd- aft
Si de- by- si de wi th
wing and reduced
di amc ter-
Side- by- aide with
wing and reduced
peripheral speed
Single- rotor with
wing and reduced
diameter
Si n gl e- ro to r . wt th
wing knd reduced
peripheral speed
U%-I
tl4
0.253
0.284
0.253
0.306
0.271
0.315
0.281
301
304
280
332
321
325
317
Ai
A/ mlB
K
to
-5
0(47.7)
0.456
0.527
0.694
0.356
0.348
0.465
0.44B
6.6
5.76
2.61
10.03
9.02
7.59
7.06
7.66
246
9.38
244
11.65
185
7.21
280
-6.24
275
— 9.17
— 7.89
268
261
0.127
0.144
0.169
0.134
0.121
0.145
0.133
V Ga
1 "
Ihov
Side- by- aide
0.264
306
0.439
7.19
— 7.66
252
0.129
Single- rotor
0.264
292
29.4
0.568
4.67
— 9.38
230
0.141
Fore- and- aft
0.264
287
0.667
3.20
—11.65
203
0.160
Side- by- side with
wing and reduced
di ameter
0.264
310
28.2
0.413
7.64
— 7.21
258
0.126
Side-by-side with
wing and rediTced
peripheral speed
0.264
317
8.0
0.357
8.61
— 6.24
272
0.119
Single- rotor with
wing and reduced
diameter
0.264
300
47.7
0.655
4.88
-9.17
235
0.138
Single- rotor with
wing and reduced
peripheral speed
0.264
309
23.3
0.477
6.16
-7.89
251
0.12f
361
The fuel supply required for flight over a given range at the cruising /337
power of the engine was found by the formula
G, _
IILC """"
\,\LCe -77^
Ga
n —
I.ILC, ^
"z
a
Kr.
^fi^fkra
(4.30)
The coefficient 1.1 is introduced to account for the fuel supply for navi-
gation and fuel consimiption in transitional regimes for a height of H = 0. Here,
Gf = -— — was calculated for L = 500 km at C^ = 0.32 kg/hp hr.
G
Table 3.12 indicates that, under equal conditions in hovering flight
(Nt.o "= Nhov)» "the fore-and-aft helicopter has the worst flying qualities in
forward flight: The rate of climb and cruising speed are appreciably smaller,
the fuel requirement is greater, the rate of descent in an autorotation regime
is greater, and -r— ^ — = 0.69, i.e., continuation of horizontal flight is pos-
Nt.
sible only if not more than one of the three engines fails.
To improve the flight characteristics, the fore-and-aft helicopter should
have a more powerful engine: N^.^ > Nhov*
The side-by-side helicopter has the best flight data.
The qualitative difference in flight data of helicopters of different con-
figurations in forward flight also remains for helicopters of equal available
horsepower per unit weight (see the second part of Table 3*12), and also if we
take into account that helicopters of different configurations have a some^what
differing flying weight at an identical lift capacity. At equal available power
per unit weight for single-rotor and two-rotor helicopters, the former can hover
only on a ground cushion, with its minimum speed outside the ground cushion
being about 30 km/hr.
By installing a wing on a single-rotor helicopter to reduce the rotor dia-
meter by 11^ and increasing the engine power also by 11^ to ensure hovering out-
side the ground effect, the flight characteristics of a helicopter in forward
flight are improved: The maximum and cruising speeds increase by 20 km/hr and
the- rate of climb by about 2 m/sec. Without an increase in engine power, the
characteristics of a helicopter in hovering flight deteriorate, but in forward
flight they change negligibly: The maximum and cruising speeds increase by
5-8 km/hr and the relative fuel feed decreases by yfo. When a wing is in-
stalled without changing the rotor diameter but with decreasing the peripheral
speed, the flight characteristics of the helicopter improve both in hovering and
in forward flight: The maximum and cruising speeds increase by 15 - 20 km/hr,
and the relative fuel feed decreases by 8 - ^%.
As noted above, the dynamic ceiling of a helicopter with a wing decreases.
362
h- Calculation of a Helicopter with a Tractor Propeller
■ty Pt.p, i.e., the drag viill be equal to -^ + Qpar - Pt.p
When calculating a helicopter with a tractor propeller (jet engine) de-
veloping a thrust Pt.p, the drag of the helicopter must be reduced by the quanti-
_G_
K
The tractor propeller requires an amount of power determinable by the
expression
Therefore, the required power of a helicopter -with a tractor propeller /338
is equal to
or
*•/ 270ET1
K ' a a
(4.33)
In calculations using eqs.(4.32) or (4.33) we must assign the value of P^.p
or Nt. p ; in so doing we must bear in mind that in steady horizontal flight the
drag of the helicopter cannot be negative; consequently, the following condition
should be satisfied:
Accordingly, in a climbing or descending regime this condition takes the
form
^<^+-^^+^. (4.35)
Such an additional term in formulas for Nh.f as in eq.(4.33) appears when-
ever tractor propellers are installed on helicopters of any configuration.
The flight data of a helicopter with a tractor propeller are determined in
the same sequence as one without tractor propellers.
Let us estimate how much the required power and maximum rate of climb may
vary when a tractor propeller is installed on a helicopter.
For Mo = 0.60 and average values of ty at high flying speeds (V ~ 0.35),
the efficiency T] of the rotor can be considered equal to 0,87. Having taken
? = 0.91 for a two-rotor helicopter and Tlt.p = 0.78, §t.p ^ 0.97 for the tractor
propeller, we find the value in .parentheses in eq.(4.33): 1 - - * — — =
= -0.05. 0.97 >^ 0.78
363
If the tractor propeller completely overcomes the helicopter drag, i.e., if
J^-.
K
(4.36)
then the increase in required power amounts to ^%, but if it overcomes this drag
only by half, then the increase in required power amounts to 2.5%- Consequently,
the losses are small.
For Me = 0.7 and V = 0.35 - 0.4, when T] « 0.85 - 0.8, there can be a
1-3^ gain in required power when a tractor propeller is used.
In a maximum rate-of-clirab regime for Mo ==• 0.6 - 0.7, V = 0.2, and at
average values of ty , the efficiency can be considered equal to at least T] =
= 0.87. Consequently, if it is possible to obtain a very high value of tractor
propeller efficiency (Tlt,p = 0.78) in a maximum rate-of-clin±i regime, then the
value in parentheses in eq.(4.33) is equal to -0.05. For a side-by-side heli-
copter, more than half of the available helicopter power is consumed in a maxi-
miom rate-of-climb regime for producing vertical speed. Consequently, when all
power is delivered to the tractor propeller, the total thrust of the helicopter
decreases by 5^, whereas the excess of thrust used for climbing decreases by
10^. The rate-of-clin±i loss will be about 1 m/sec. At T] = 0.9, we have 1]^, p =
= 0.7, and when half of the available power is supplied to tractor propellers /339
the rate-of-climb loss will amount to 2Q%, or about 2 m/sec. General considera-
tions on when the installation of a tractor propeller or other propeller on a
helicopter is expedient or necessary
are given at the end of Section 7,
Kt]1[ I I I J_J I I I I I I T^ I ] Chapter II.
5. Comparison of Helicopter and Airplane
At identical flying weight of heli-
copter and airplane, the relation D =
= Lw is approximately satisfied; this
relationship is determined by design
advisability of the size of wings and
rotors. A comparison of the parameters
of regular aircraft and helicopters
shows that a wing has an aspect ratio
larger by a factor of 7 - 9 and an
equally larger load per square meter of
area. Consequently, ( j =
I'M aire
Fig. 3. 50 Comparison of the Product
of Aircraft Performance and Effi-
ciency for Regular Aircraft and
Helicopter.
const. This means that a given aircraft and helicopter have
an approximately equal induced drag at the same flying speed.
Provided that D = Lh, the dimensionless coefficients of an airplane and
helicopter are connected by the relation
364
G-
'^i/. Y^'-^-^Vf-C*"^)'^.
c,, =
tyO JlXj,
(4.37)
For a comparison, let us take the following data of a helicopter and an air-
plane. For a helicopter, a - 0.091; ty = 0.15, "c, ^ OtQfyi'i, For an airplane,
the -wing characteristics are taken from data of wing exposirre to propeller slip-
stream of a modern low-speed transport aircraft. A rectangular wing with a slat
and double slotted flap was tested. The wing aspect ratio was X» = 9. The
parasite drag coefficient c,^ ^ of the airplane, based on wing area, was taken
to be equal to 0.025 [Sc^S of the airplane is approximately one half that of a
helicopter:
(Sc,S),ir,
rot
copter, Cx = 0.0075].
aire
■'aire
ro t
0.025
TT JU.
4
0.0036 and for the heli-
TABLE 3.13
7340
Slat
Slot
Defl ection
of Fl ap
K„
" aire,
^airc.'^t.,
i\.p =0-7)
'^airc.'^t.p
(tl^p=0.85)
Retracted
Overl apped
Open
20 40 50
0.4
0.8
1.2
1.45
2
2.4
0.0185
0.0439
0.0873
0.1175
0.232
0.417
21.6
18.25
13.75
12.35
8.63
5.77
9.2
11.6
10.65
10.2
7.8
5.44
6.44
8.12
7.45
7.15
5.46
3.8
7.82
9.86
9.05
8.67
6.62
4.62
K.n^
0.479
0.34
0.277
0.252
0.214
0.196
—
9.3
8.2
7.3
5.9
5.12
—
0.875
0.915
0.93
0.965
0.975
—
5.75
6.03
5.77
5.1
4.66.
—
4.43
4.85
4.72
4.33
4.00
Moved
Forward
Open
2.55
0.451
5.6
5.36
3.75
4.55
0.19
4.19
0.98
4.5
3.88
SO
2.87
0.582
4.93
4.72
3.31
4.02
0.179
4.45
1.0
4.11
3.62
For the selected values of Cy^ we find, in Table 3.13, c^^^, K«, Y^%ira»
Kairc'Ht.p. Foi" these same values of Cy , the values of V, K, T], Kh, and KuT] are
365
determined for the helicopter. Since the helicopter has additional engine power
losses, the product Ki,T15, where % = 0.88, is calculated for a single-rotor heli-
copter with a tail rotor. The graph in Fig. 3. 50 is constructed from the data of
Table 3.13.
A comparison of wing and rotor performance shows that, in the examined
example, the wing without mechanization, at all Cy^^, has a performance greater
by a factor of 2 - 1,7 than the rotor. The wing with mechanization has a 1^6%
greater performance at Cy = 2.0 (at this c„, the minim.um flying speed corre-
sponds to V„i„ = 0.214), whereas at (cy )aax "= 2.87 (Vnm = 0.18), the wing
performance is only 10^ higher than that of the rotor.
It follows from Table 3.13 and Fig. 3. 50 that the maximum value of ^e,iTe\.v
^'^ ^t.p = 0.7 is by a factor of 1.75 greater than Kj,1i1§. The fuel consumption
of the airplane per kilometer is less by the same factor than that of the heli-
copter (at equal specific consumptions of the engine). The speeds corresponding
to maxima of the product of the craft performance and efficiency is by a factor
of 1.2 greater for the airplane (actually, airplanes fly the range at a greater
speed and with a performance less than maximum) .
On comparing a helicopter and an airplane at equal flying speeds, it will
be found that at speeds reached by an airplane without the use of wing mechaniza-
tion (V s 0.3 - 0.25), (K'n)air8 is by a factor of 1.5 - 2 greater than (K115)h.
At V > 0.43, flow separation from the rotor blade begins at the helicopter. At
low flying speeds, reached by an airplane with the use of powerful mechanization,
(K'Tl)airo is less than (KTlg)^ owing to the large profile drag of the wing. Thus,
it is aerodynamically less expedient to use such an airplane with its low at-
tainable speed for long flights; a helicopter is then preferable.
Table 3.13 shows that, at equal flying weight and at D = L^, the minimum /341
speed of the airplane, determined by the quantity (cy^)nax = 2.87, will be V =
= 0.18. A low flydng speed can correspond to this value of V only if the air-
plane has a small wing loading. The minimum speed of the helicopter is deter-
mined by the available engine power and is usually equal to zero, whereas_when
the helicopter is overloaded it will not exceed a value corresponding to V =
= 0.05.
6. Power of Front and Tail Rotors in a H elicopter of
Fore-and-Aft Configuration
An expression was derived in Subsection 2 for determining helicopter per-
formance and total required power of both rotors. However, the tail rotor usu-
ally requires substantially greater power than the front rotor (by a factor of
•1.5 and more). Let us derive an expression for determining the power required
by each rotor separately.
First, let us find the propulsive forces of the rotors. They are not
identical, since various rotors may have different lifts and different angles of
attack.
366
According to eq.(4«8), the relation between ax and 0^3 is equal to
Oj = a, — ^Ont, -j- Ae„4 ,
where Ao^-oto is determined by eq.(4.14) with consideration of eq.(4«lS).
Proceeding Itom the approximate expression for X:X = Y{a + a-x) , we find
''1
From the condition of equilibriimi of forces acting in the direction of
motion (Fig. 3. 4), we obtain the following equality:
x,+x^^r^^a„,,-]-Qfar=o. (4.39)
In eq.(4.39), Qpar ^°^ ^ helicopter with a wing and tractor propeller re-
presents the sum Qpar + ^w + 'i,, l^0!„ - Pt.p*
From eqs.(4.3S) and (4.39), we obtain
^1 = yr Qp^r- ^ AB^t, ; ( 4 . 40)
1 + ^ 1 + -^ (4.41)
Equations (4.40) and (4.41) indicate that vee-ing of the rotors by an angle
Asrot redistributes the propulsive forces of the rotors, thus influencing the
power required by the rotors. Owing to downwash of the tail rotor, its propul-
sive force is greater (more negative) by an amount of YsAotots* ^'^ ^1 ^ Yg , the
front and tail rotors do not furnish an equal share of parasite drag.
Substituting Xi and Xg into the expression for calculating the required /342
power
j,j YiV / \ X^\ GV lYx X\ / , . „N
we obtain
367
jg.
It is easy to demonstrate that, on adding the expressions in parentheses in
eqs,(4.43) and (4.44)> tihe sum will coincide with eq.(4-13) for the case of a
fore-and-aft helicopter. The angle of vee-ing of the rotors Asrot ^.nd the
Yi
ratio at Tli = Tig will not influence the total power of the helicopter. At
Yi = Ys and Aerot == Affrots* ^^^ power of both rotors is identical. It is
necessary to note, however, that' the longitudinal stability of the helicopter
deteriorates when Srotg > 0.
7. Retraction of Landing Gear on Helicopters
It is known that helicopters have a parasite drag about twice that of
regular aircraft.
This can be attributed to the specific configuration of a helicopter, the
presence of a cabane and large hub of the rotor^ tail boom with a tail rotor
placed high, and also to the necessity of loading and unloading in hovering
flight and maintenance without the use of an airdrome. Therefore, various
hoisting devices, numerous railings and hatches, movable doors, blisters, etc.
are often installed on the outside of a helicopter. On the other hand, the
weight coefficient which is lower than that of a regular aircraft necessitates
a careful approach to any measures that reduce the parasite drag but increase
the structural weight.
Below, we will estimate the expediency of installing a retractable landing
gear on a helicopter from the viewpoint of its load-carrying capacity.
Retraction of the landing gear reduces the parasite drag of a helicopter by
20 - 25%- Figure 3.51 shows the graph of the ratio — ^ — for two values of
Cjt : 0.0075 and (by 23% less) 0.0056. The graph also contains the quantity
— 7^ » 0.7 — f-i — -• It is obvious that, with retraction of the landing gear,
the cruising speed of the helicopter increases from Vg^u = 253 km/hr (Mfj =
= 0.207) to Vcru = 269 km/hr (Mn = 0.22), i.e., by 6%. The required fuel sup-
ply will decrease by this same amount, while the range will remain unchanged.
If the cruising speed is retained, then the required power diminishes by 9%
368
( -j=i — = 0.000555 in place of O.OOO605). Since the specific fuel consumption of
turboprop engines greatly increases, upon a decrease in engine power, a change in
power - as shown by calculations - will result in a change in fuel consimiption
smaller by a factor of about 1,5. Consequently, we arrive at the same figure:
The required fuel supply decreases by 6^.
Now we can calculate that portion of the structural weight increase by /343
retraction of the landing gear which is compensated by a decrease in fuel re-
quirement. Thus, an increase in structu-
ral weight by 1% of the takeoff weight
will be compensated at a fuel requirement
equal to 17^ of the takeoff weight, i.e.,
Gf = 0.17, since 6% of 17^ is 1%, A
1,5^ increase in structural weight will be
compensated ■when Gf =0,25.
0.00075
0.0005
0.00025
Fig. 3. 51 Ratio Nh . ^ /Ga of Heli-
copter for Two Values of Para-
site Drag Coefficient, as a
Function of Mfi .
The normal fuel supply of modern
turboprop helicopters is about 15^ of the
takeoff weight, with a maximimi of 20-25^.
It is obvious that installation of a re-
tractable landing gear on modern heli-
copters is expedient if the increase in
structiiral weight does not exceed 1 - 1.5^
of the takeoff weight. In so doing, how-
ever, the maximum load-lifting capacity
of the helicopter decreases in flights
with a smaller fuel supply.
A 6% decrease of fuel supply and an
equal increase in cruising speed lead to cheaper hauling on helicopters, which
should also be taken into account by the designer vjhen attacking the problem of
landing-gear retraction.
It should be pointed out that, on airplanes with a higher performance of
the lifting system, a decrease in parasite drag will lead to a greater decrease
in fuel consumption. Furthermore, regular aircraft have greater relative fuel
supplies Gf , for which reason retraction of the landing gear on airplanes has
become advantageous at cruising speeds lower than those presently used for heli-
copters.
Section 5. Aerodynamic Calculation of a Helicopter by the Power Method
In an aerodynamic design of a helicopter by the power method, the condition
of power balance in steady helicopter flight is used: The power supplied to the
rotor is equal to the simi of all power losses. Thus, having determined all
losses of power - both of the profile and induced type - produced in overcoming
the parasite drag of the nonlifting parts as well as the helicopter weight com-
ponent in climbing, we find the power which must be supplied to the rotor.
The formulas for determining the torque coefficient of a rotor, derived in
369
Section 3, Chapter II, express the condition of the power balance. The same
Section contains formulas and graphs for calculating the profile and induced
power losses for a rotor.
It is general practice to determine all power losses approximately so as /344
to simplify the calculations; therefore, the aerodynamic calculations of a heli-
copter by the power method constitutes an approximate method.
1. Determination of Required Power in Horizontal
Helicopter Flight
The required power of a helicopter is equal to the sum of the profile and
induced losses at the rotor and the loss due to overcoming the parasite drag of
nonlifting parts of the helicopter
^v =^^'- +^«^ +^^- (5.1)
The profile power loss coefficient mpr is determined from special graphs,
or by the approximate formula (3.75) from Chapter II:
m,r=-\-'^.p,,il + 5V')P-\-Ant,,. (5.2)
In dimensional form, the profile losses of a rotor are calculated by the
formula
^^^-^f-^p^^(^^r^^ (5.3)
where § is the engine power utilization factor.
The induced power loss coefficient is determined by means of eq.(3.83)
from Chapter II
m.^^^_^± =0.285-^, (5.4)
vAiile, in dimensional form, the induced losses of a rotor are determined by
The induced losses can also be represented as the product of the force of
the induced drag of the rotor and the flying speed, or as the product of rotor
lift and average downwash angle in the rotor plane and flying speed
N. = '^ ^ind^^^ K(aa)K
'"d 52 755 B2 755 ■ . (5.6;
The average downwash angle in the rotor plane is determined by the average
370
induced velocity of the rotor
2fb2qv 2fb2qv ' v5.7;
Aa=— = — ^ . (5.8)
V 2FB2QV2 ^^'°^
It is obvious that eq.(5.5) is also obtained from eqs.(5.6) and (5.8).
It follows from eq.(5.5) that the induced power losses are directly pro-
portional to the square of flying weight, referred to the effective linear di-
mension BD (i.e., the span of the lifting system determines the volume of air
flowing through the rotor). The induced losses are inversely proportional to
flying speed and air density.
Consequently, upon an increase in helicopter weight without a proportional
increase in rotor diameter, the induced losses increase with respect to a /345
n
quadratic relation. If the load per unit rotor disk area p = -^ is retained,
then the induced losses will be directly proportional to the flying weight and
the ratio — -=; will remain unchanged. However, since increasing the helicopter
tonnage causes p to increase (for decreasing the relative weight of the rotor),
Nina
the ratio — ^ is greater for heavy helicopters than for light ones.
For multirotor helicopters, the induced power losses are determined as the
svaa. of the product of the type of eq.(5.6), taken for all elements of the lift-
ing system:
^w=4"S"l'^^"- (5.9)
In eq.(5.9), the downwash angles Aa are equal to the sum of all downwash
angles for each element of the lifting system: the downwash angle due to self-
induction defined by eq.(5.8) and the downwash angles due to interference*,
whose expressions are given in Sections 1 and 4.
As a typical example, let us develop eq.(5.9) for a fore-and-aft helicopter,
using eqs.(4.14) and (4*18) for the downwash angle due to interference:
^M=-^ ^-(l^iAai + KjAaj);
.Aai =
2/='B2qK2 '
■5S- For the terms containing the downwash angles due to interference, we can. take
$/B^ = 1.
371
^°^^ OP-L\.o + 2\
2rB2QV2 ' <=» 2FB2QV2 •
Having substituted Acci and Aofe into the expression for Nma, we obtain
If the rotors have no excess, then the graph of k^o in Fig. 3. 8 furnishes
Koo = 1 and
^>n,=-:^,,^&^iy^+y2y- '"" °'
m2B'iQVF ^ ' ' 75ilB4£ i/A£)2
The expression for Nm^ shows that, at Koq = 1, the quantity Ni^^ does not
depend on the distribution of helicopter weight between the front and tail
rotors and is detennined only by the sum of lifts Yi + Tg = G. Displacement of
the lifting elements along the direction of flight does not influence the quanti-
"ty Ni„4, so that the expressions for N^a coincide for single-rotor and fore-
and-aft helicopters.
However, it must be borne in mind that, for a two-rotor helicopter of fore-
and-aft configuration, the flying weight is equal to the thrust of the two rotors
and that, at identical load on the rotor disk area p, the ratio G/D is twice
that of a single-rotor helicopter. Therefore, as already indicated in Chapter I,
Njni is by a factor of 4 greater for a fore-and-aft helicopter than for a single-
rotor tjrpe, and the ratio — tt^— is twice as large. This explains why fore-and-
G
aft helicopters have poorer flying characteristics in horizontal flight and -vAiy
the fl3ring characteristics deteriorate more noticeably upon an increase in /3h^
flying weight.
If K,„ ^ 1, then N,„, = -i^i ^ ^t""" •
For the side-by-side helicopter, the effective transverse dimension, i.e.,
the span of the system, increases with increasing flying weight, which is ex-
pressed by the fact that Kg . s < 0. At Kj . g = (with the rotors spaced far
apart), — p"'^- is the same for the side-by-side helicopter as for the single-
rotor helicopter; at Kg. a < 0, the ratio — pr^— of the former is lower.
G
Losses for overcoming the parasite drag of the nonlifting parts of a heli-
. copter are determined by the formula
N^=S^=-^^V^^^^^- V^^ (5.10)
'^ 765 16-755 12005
or, in dimensionless form,
372
'"/«•= -<,^^V^, (5.11)
where
S = ~'-^^- ^5.12)
When calculating a helicopter with a wing, we will refer the wing drag to
the parasite drag of the helicopter, i.e.,
2^^ = (2-.5)^+('^.. +^..Aa,)5,, (5.13)
where c,^ is the parasite drag coefficient of an isolated wing; Aa^ is the down-
wash angle of the wing due to interference of the rotors.
The interference of the wing with the rotors should be taken into account
when determining the total rotor downwash angles for calculating Nm^ by
eq.(5.9).
Thus, the required engine power of a helicopter, in conformity with
eqs.(5.l), (5.3), (5.5), and (5.10), is equal to
N. =Np^ +-' 91 iSff^ ^3. ( 5 .14)
If the helicopter has a propeller, then the power balance is expressed in
the form
The aerodynamic calculations can be performed in dimensionless form. In
this case, the coefficient of required torque is determined, in conformity with
eqs.(5.4) and (5.11), by the expression
'"V ^ '"'"" + °'^^^ "^ " S "^ • ^5-^5)
As a dimensionless form of calculation, convenient - for example - for
the method of powers, is equal to
comparative calculations, we can determine the ratio \* ' which, when using
Ga
Nl
Ga~ 75£(<3,aMg) 'V
where
t^oMl= ^—
'"f.x^'M?,
(5.16)
/347
(5.17)
373
..3 '^Oy''M.lY
(5.18)
(5.19)
At given p = -ri-, Cx = — s * height, and speed, the value of the products
tyQMo and t^^^ jCrM© does not change upon a change in the rotor parameters uiR and-
CT. Consequently, when studying the effect of rotor parameters on the magnitude
of required power of the helicopter, eqs.(5.l6) and (5.18) are transformed into
frif oM3= consta ■i-mpr<^M.l,
As an example of the aerodynamic calculation by the power method, let us
determine the required power of the Mi-4 helicopter. The helicopter data were
given in Section 3. The calculation is performed by means of eq.(5.14) in
Table 3.14. For simplicity, the profile losses were determined from the graphs
in Figs. 2.63 - 2.66, using eq.(6.10) of Chapter II for converting t,; ; we can
disregard the differences in blade profiles for Mo = 0.6.
TABLE 3.14
G = 7200 /t^. ; a = 0,063; « /? = 196 /w/sec! S = 0.84;
/=• = 346 m 2; Mo = 0.6; ty = 0.138— ; N^r = 163-103 m^^A;
N,-„^ = mO — ; Npar=
2c^
1010
V3A;
//=1000/»,
<y =0.152; A =
= 0,907
V
0.15
0.20
0.25
0.30
V, Mm/hr
106
141
176
212
V, m/szc
29.4
39.2
49
58.8
V
4.5
2.75
-3
Cx
0.0088
0.0089
0.0092
0.0096
^CxS
3.04
3.08
3.18
3.32
>
—0.00314
—0.00565
—0.00915
—0.0137
0.0046
—0.00135
—0.0063
—0.0117
nipf
•0.00286
0.00306
0.0034
0.0037
Npr
420
450
500
545
Nind
338
253
203
168
Npar
70
165
340
610
V
828
868
1043
1323
374
2. Determination of Helic o pter Pe rformance Data /348
The dependence of required power on flying speed is found by means of the
formulas given in Subsection 1. Maximum and minimum speeds, maximtmi range and
endurance, cruising and economic speeds are then determined by the method de-
scribed in Section 3.
During ascent, the propulsive force of the rotor increases by an amount
equal to the projection of the helicopter weight onto the flight direction
G sin Qfi.p. Consequently, the engine power of the helicopter increases by an
amount of
A^..c = 7^Gsin9,i,l/=^GK, (5.20)
while the total power of the engine is equal to
N^Npr^Nind^Np^+N^^. (5.21)
Here, N^g o represents the variation in potential energy of the helicopter
upon a change in its flight altitude.
The components Np^, Ni„a, and N^^r o^i vertical ascent and in horizontal
flight differ somewhat in magnitude. However, for approximate calculations we
can disregard this, and under this assumption eq.(5.21) can be represented in
the form
It is obvious that the maximum vertical rate of ascent of the helicopter is
y^_jm,-s_^^^t:l,,^^ (5.23)
V
max
The discrepancy between eqs.( 5. 23) and (4. 24) can be explained by the as-
sumption that Njnd 3-i^d Npr are equal in horizontal flight and vertical ascent.
Equation (4»24) gives a more correct result. At flight altitudes where the
rotor lift coefficient is less than the maximum value permissible with respect
to flow separation, we can take, in conformity with the graphs in Figs. 2.166
and 2.167, the average value of the propulsive efficiency as equal to 0.95 and
determine V„ by the formula:
ym ax *'
"max
0.95-75(Ar^ ,, -^*/n,m)S (5.24)
At high flight altitudes in a climbing regime, where ty ~ ty^ , the speed
Vy should be determined from eq.(4.24).
B ax
375
3. Relation between Np,, Nj^ij, and Npar during Horizontal
Flight of a Single-Rotor Hel icppt er
It is of interest to examine the relation between individual components of
the required power of a helicopter. Since the helicopter parameters determining
N
pr» ^ind*
and N ar depend on the gross weight of the helicopter, we will give
data for helicopters of different weight classes,
Helicopters of different weight classes have a maximum weight coefficient
at a different load per square meter of rotor disk area p and correspondingly
have different peripheral speeds and solidity ratios, since the lift coefficient
t limited in value by flow separation should be within 0.23 - 0.27 at the dy-
namic ceiling and 0.13 - 0.17 near the ground.
^I'P/Ms
l^y - ■ — /, ight helicopter
, 7~ — — — — AfediuB helicopte
op t
Mg=0.7 \ helicopter
0.3 Mfl
Fig. 3. 52 Quantity Nh.f/G and Relation
between Components of Required Power of
Helicopters of Different Weight Classes.
376
Let us assume that the charac-
teristic parameters for a light
helicopter with an all-up /349
weight to G = 3000 kg are:
Mo = 0.55, o- = 0.05; for a
mediiom helicopter G =
7000 - 14,000 kg; Mo = 0.6, o =
= 0.07; for a heavy helicopter:
Mo = 0.65 - 0.7, CT = 0.09.
The quantity Ec^S of the
helicopter referred to gross
weight decreases upon an in-
crease in weight owing to the
relative decrease in the overall
size of the helicopter (so-
called "scale effect"). How-
ever, with an increase in G of a
helicopter the value of p will},
increase, while the parasite
drag coefficient referred to the
rotor area, Cj^ =
G
will
change little for helicopters
of different weight classes.
Let us take it to be equal to
0.0085 for light and medium
helicopters and 0.0075 for
heavy helicopters.
For calculations at V ^ 0,
we use eqs.(5.4), (5-ll),
(5.12), and (5.16). The coeffi-
cient mpr is found from the
graph in Figs. 2. 63 - 2.70 as a
function of the coefficients ty
and tjt . At V = 0, we use
eq.(8.27) and the graph in
Fig. 2. 174.
The calculation results are shown in Fig. 3. 52. Owing to an increase in p,
a. Mo, the quantity — ^f-^ — in hovering flight is much greater for heavy heli-
G
copters than for light helicopters.
In forward flight, eq.(5.l6) can be written in the form /350
° '"' [ '^ -f P..AV "" ' ) (5.25)
Nh f
This expression indicates that, at high flying speeds, the value of — ^ ,
despite the increase in p and Mo, is lower for heavy helicopters than for light
ECxS
ones owing to a decrease in the ratio - — .
If the available power of helicopters is equal to the required power in
hovering flight, then the average value of the majclmum speeds of helicopters of
different weight classes is equal to 210 km/hr (Mfi = 0.1?), 260 kmAir (Mf i =
= 0.21), and 310 km/hr (Mn = 0.25).
The graph shows that the profile power losses, in percentage of the power of
horizontal flight, are 22 - 2?^ at V = and 50^ at average flying speeds, while
they are k5% for light and medium helicopters and 55^ and more for heavy heli-
copters at Vnaic* I't will be recalled that the graphs of mp^ in Figs. 2.63 - 2.70
pertain to a rotor of average blade manufacturing quality and that Cxp of the
profile increased by Ac^p = 0.002.
The induced power losses amount to 73 - 7&% in hovering flight, 1+0% at
average flying speeds, and only 13^ at maximum speed.
Losses due to parasite drag amount to 15 - 10^ at average flying speeds and
to 40 - 35^ at maximum speed.
Thus, it turns out that, although helicopters of different weight classes
differ in speed range, in load per square meter of rotor disk area, in peri-
pheral speed, and in relative parasite drag, the power losses in fractions of
the required power show a distribution that is practically the same at corre-
sponding speeds.
The above data permit an approximate estimate as to the degree of variation
in required power of a helicopter on introduction of various modifications in
the helicopter design. For example, an improvement in blade finish may cause its
profile drag to decrease by 20^; consequently, the required power of the heli-
copter will decrease by 10^ at medium and high speeds. In hovering flight, the
required power diminishes by 5^, which is very substantial since, in this case,
the relative efficiency of the rotor increases by a like quantity while the
maximum thrust of the rotor increases by — k— ^ ^ = 3*3^ [the coefficient -75— is
377
obtained in accordance -with eq.(8.34) of Chapt.U].
Upon a change in blade shape, the induced losses of the rotor may vary
within several percent. It is obvious that this substantially affects the maxi-
raum thrust of the rotor in hovering flight but practically causes no change in
the required power at high flying speeds. The change in blade shape at large Mo
significantly changes the rotor profile losses (see Sect .3, Chapt.U).
A 25% decrease in parasite drag of a helicopter leads to a 3% decrease in
required power at medium flying speeds and to a 10^ decrease close to maximum
speed; this yields an increase in maximum speed by 15 - 20 km/hr.
378
CHAPTER IV 7351
ROTOR FLUTTER
The phenomenon of rotor flutter has been a persistent companion of the de-
velopment of helicopter construction. Numerous cases are known of the occur-
rence of flutter in experimental helicopters during their first ground test or
during flight tests. Cases of the appearance of flutter have been observed also
during operation of helicopters that had already undergone all test stages.
Rotor flutter has been the cause of a number of accidents.
The greatest number of cases of flutter was observed at a time when this
phenomenon had not yet been adequately studied and due attention had not yet
been given to its investigation. At present, flutter has been studied in great
detail, and there are numerous means for completely preventing its appearance.
However, the helicopter designer must keep constant track of the rotor para-
meters and hold them to limits that ensure the necessary safety margin before
onset of flutter. However, these parameters vary constantly with design and
technological modifications made in designing and plant testing of a helicopter
and during its series production. Such variations continue even when the heli-
copter has been placed in service. This is due to various circumstances. The
most common case is deterioration of the individual blade balance, either due to
penetration of moisture into the blade or due to its increase in weight during
repair.
Experience shows that even the slightest letting up in control of the rotor
parameters will immediately cause appearance of flutter. This is primarily ex-
plained by the fact that the designer strives to reduce the margin before onset
of flutter to a minimum since the expenditures produced by an increase in rotor
weight are generally proportional to the magnitude of this margin. Its increase
requires a corresponding increase in blade weight or in weight of the structiiral
elements of the rotor control system.
As a result, the most economic design of a helicopter keeps these para-
meters at the minimum level allowable by the flutter limits. Even their slightest
variation produced by some unforeseen happening may lead to flutter. The blade
parameters, at all times, are kept close to the flutter limit.
This circumstance necessitated taking reliable measures to keep the flutter
characteristics of a rotor within limits that would ensure prescribed margins /352
before onset of flutter, which, as a rule, are rigorously standardized. These
measures should be enforced both during production and service of the helicopter.
In addition, each helicopter must be subjected to special ground tests to check
for flutter in the final inspection. Experience gained in mass use of helicop-
ters confirms the reliability of this inspection system. We can consider that,
at present, conditions have been created that preclude the possibility of acci-
dents owing to flutter. Actually, cases of unforeseen occurrence of flutter
have almost completely stopped.
379
The achievement of the present favorable state as regards flutter was pre-
ceded by extensive theoretical and experimental investigations.
Valuable contributions to the development of the theory of flutter were
made by P.M.Riz, L.N.Grodko, V.D.Il'ichev, M.S.Galkin, A.I.Pozhalostin, F.L.
Zarzhevskaya, M .E .Lipskaya, V Jl.Pchelkin, and many other engineers. Numerous
papers by foreign authors are also well known [see (Ref.39 - 42)].
Results of great importance for the development of the theory were obtained
in flight tests on flutter carried out by S.B.Bren and A.A.Dokuchayev and per-
formed by the pilot V.V.Vinitskiy.
Many highly useful results were obtained by L.S.Popov, B.A.Kirshteyn,
N.V.Lebedev, and B.BJ^artynov in tests of dynamically similar models.
All this work led to rather clear and distinct concepts concerning the phe-
nomenon of flutter which permitted developing new blades with the necessary
parameter margins, without additional modifications after tests, as had often
been necessary before. However, for this it was necessary, in designing the
blade, to perform numerous rather laborious calculations. This Chapter will be
devoted mainly to an account of the method of these calculations.
In writing this C3iapter F.L. Zarzhevskaya was of considerable help to the
author, for which the author extends his gratitute.
Section 1. Basic Assumptions and Characteristics of an
Approach to Flutter Calculation
1. Bending and Torsional Vibrations of the Blade .
Possible Cases of Stability Loss
The theory of rotor flutter is developed on the basis of an investigation
of bending and torsional vibrations of blades during their rotation in air.
When solving the problem of bending and torsional vibrations of blades in
air, the designer is interested primarily in two qualitatively different prob-
lems. The first of them reduces to a determination of steady bending and tor-
sional vibrations of the blade, occurring in all helicopter flight regimes.
This problem requrres the development of special calculation methods which are
a further development of calculation methods for forced vibrations of a blade
and should, in particular, answer the problem of the effect of torsional de-
formations of a blade on its bending vibrations and, accordingly, on the magni-
tude of variable stresses from blade bending. The second question is associ-
ated with a determination of the stability of blade motion. Usually, purely
bending vibrations of blades are stable. Loss of their stability occurs only
in flow-separation regimes.
In studying bending and torsional vibrations we find that, at certain rotor
parameters, there is a loss of stability of motion of blades which leads to
flutter or divergence. The phenomenon in which blades undergo oscillatory in-
stability is called flutter, whereas the phenomenon of aperiodic instability of
380
blade motion is called divergence. The most common of these two phenomena /^^'i
in practice is rotor, flutter. Therefore, when examining bending and torsional
vibrations of a blade, the designer is more interested in the conditions leading
to flutter.
2. Effect of Blade Attachment to Hub and the Possibility
of TheoreticaT In vestigation of Flutter of an
Isolated Blade
The results of calculating flutter largely depend on the design configura-
tion of the rotor and primarily on the conditions of blade attachment to the
root, i.e., on hub design and rotor control system. The characteristics of
blade attachment influence the boundary conditions of the problem and hence the
design formulas for determining flutter parameters.
The most common type of rotor with individual hinge ' attachment of each
blade to the hub, with the control exercised over an automatic pitch control
mechanism, will be examined below when presenting the method of calculating
flutter. For rotors with a rigid and universal joint attachment of the blade to
the hub or with some other type of control, the approach to flutter calculation
remains the same. However, the conditions under which flutter occurs may change
extensively.
Flutter is greatly influenced by the design of the system controlling the
angle of blade setting and, primarily, by the design of the automatic pitch
control. The automatic pitch control couples the oscillations of the different
rotor blades. Thus, as soon as this couple becomes sufficiently strong - and
this generally takes place on real helicopters - it is impossible to investigate
the flutter of an isolated blade. It is then necessary to study the flutter of
the entire rotor a.s a whole.
In all practical cases, there occurs only flutter of the entire rotor as a
whole, when each advancing blade of the rotor duplicates the motion of the re-
treating blade with some lag. Flutter of a single blade has never been noted.
However, in many cases the investigation of flutter of a rotor as a whole
can be reduced to calculation of the vibrations of an isolated blade. Therefore,
calculation of the flutter of an isolated blade often furnishes a sufficiently
comprehensive answer so that we can frequently restrict ourselves to this result
in practice. In so doing, however, it is important to properly prescribe the
stiffness of the isolated blade control. This question will be taken up in
greater detail in Section 4»
3. Different^. Types of Fl utter Differing with Respect to
Blade Vibration. Flapping^ and^ending Flutter
The problem of determining the conditions for occurrence of flutter is
solved usually by means of differential equations of bending and torsional
(binary) vibrations of the blade (see Sect. 6). These equations permit obtaining
the parameters of different types of flutter which differ by the blade vibration
381
modes. The critical rpm and other parameters of flutter obtained from solving
these equations are quite complex functions of the initial rotor parameters.
Therefore, an analysis of these relations is conveniently begun with the simplest
particular case. In fact, flutter in which blade vibration in the flapping plane
occurs mainly with the fundamental vibration mode of the blade is most wide-
spread in practice. Bending strains of the blade in this case have the charac-
ter of an admixture to the vibration mode and do not determine the phenomenon. /354
Therefore, in this case all relations of interest to the designer can be ob-
tained from examination of a rotor model with blades that have absolute flexural
rigidity and execute flapping vibrations about the flapping hinges. This type
of flutter will henceforth be called "flapping flutter" in contrast to "bending
flutter", whose characteristics cannot be determined without regard for the
flexural deformations of the blade.
4. Characteristics of the Torsional Vibration Modes of
a Blade and Possible Correlated Assumptions
The relation between torsional rigidity of the blade and the rotor control
system in most modern helicopters is such that, in torsional vibrations, the
blade turns mainly as a consequence of deformations of the controls (Fig. 4.1) »
In this case, the setting angle cp
of the blade element over its
length, especially at the most
effective portion from r = 0.5 to
the blade tip, vary so insignifi-
cantly that in flutter calcula-
tions we can set, with a suffi-
cient degree of accuracy.
as
—
—
\
1
~
—
■^
=
'
V
-§t
—
— "
\}
r^
_,
,^
■^
. —
-■
r\
T
w
—
:::=
,
^
\
^
■ —
~~~
-
-
-
-
—
-
-
-
-
_
1
2
3
a
*
0.
i
0.
f
0.
7
0.
8
0.
9
1
r
q)=const.
(1.1)
The acceptance of this law
of angle distribution cp is equi-
valent to the assumption that the
blade is absolutely rigid in
torsion and executes torsional
vibrations only as a consequence
of deformations of the control.
To have this assumption lead to the smallest possible error, we will introduce
into the calculations the equivalent value of the hinge stiffness of the con-
trols which takes into account the elasticity of the blade itself.
Fig. 4.1 Typical Natural Vibration Modes
of a Blade in Torsion (the Curves Refer
to Three Different Helicopters) .
Calculations made to substantiate this assumption show that it can be suc-
cessfully used for all rotors for which the values of the angle a < 0.1+ - 0.5
(see Fig.4«l)» which probably encompasses almost all existing helicopters.
It should also be noted that the described character of the relation be-
tween torsional rigidity of the blade and its attachment causes the axis about
which the blade elements in torsional vibrations are turning to come close to
the axis of the axial hinge. Hence, the position of the axis of blade stiffness
in the examined cross section loses its significance. This circimistance permits
382
the approximate assiraiption that, in torsional vibrations, the blade elements
turn about the axial hinge.
5. Assumptions on Blade Oscillations in the Plane of Rotation
7355
There exists a definite coupling between blade vibrations in the flapping
plane and in the plane of rotation. This coupling is due to two types of forces.
The stronger is the coupling created by Coriolis
forces. The weaker is the coupling due to aero-
djmamic forces.
Let us examine in some detail the forces
coupling vibrations in the flapping plane and in
the plane of rotation.
During vibrations in the flapping plane,
Coriolis forces arise which act in the plane of
rotation
(X =- - 2,
'"yy m,
(1.2)
where
Fig. 4. 2 Coriolis Forces
Acting on a Vibrating
Blade.
y = rate of displacement of the blade ele-
ments in the flapping plane (Fig. 4. 2);
y'= angle of inclination of the blade axis
upon deflection of the blade from the
plane of rotation;
m = mass of the blade element.
During blade vibration in the plane of rota-
tion, variable Coriolis forces are set up which act in a direction close to the
direction of centrifugal forces. These forces stretch the blade and therefore
should be taken into account in differential equations of blade vibrations,
along with centrifugal forces.
The Coriolis forces acting in the direction of the blade axis can be deter-
mined by the formula
N^=~2<iixtn, . .
where x is the rate of displacement of the blade elements during vibrations of
the blade in the plane of rotation (Fig. 4. 2).
The Coriolis forces determined by eqs.(l,2) and (1.3) relate the blade vi-
brations in the flapping plane and plane of rotation.
The aerodynamic forces create an analogous coupling.
If, in the flapping plane, variable aerodynamic forces associated with a
change in the value Cy act on the blade, then the component of these forces
383
Q=OT (1.4)
will cause blade vibration in the plane of rotation [the value of i entering
eq.(l.4) determines the angle of inflow].
During blade vibration in the plane of rotation, the aerodynamic forces
acting in the flapping plane will vary as a function of any variation in the
relative velocity U.
Thus, the presence of the described couples requires that blade vibrations
in the plane of rotation be taken into account also in flutter calculations.
However, calculations and experiments show that blade vibrations in the plane of
rotation have an insignificant effect on the critical numbers of revolution of
flutter. Therefore, in all calculations of flutter, blade vibration in the
plane of rotation can be disregarded. We must also take into account that, in
the absence of thrust at the blade, when the angle of inflow § is equal to zero
(such a position is possible for an untwisted flat blade) and the blade is not
deflected from the plane of rotation so that y' = 0, the terms of the coupling /356
determined by eqs.(l.2) and (l.4) are absent. Thus, in this case there is no
coupling between vibrations in the indicated plane.
6. Determination of Aerodynamic Forces Acting on
a Vibrating T^ofile
The occurrence of diverging vibrations in flutter is caused by aerodynamic
forces acting on the blade profile. Therefore, the basis on which these aerody-
namic forces are determined is very important.
In performing practical calculations of flutter, the method of determining
aerodynamic forces based on the "steady-state hypothesis" is widely employed.
In this hypothesis, it is assumed that, during vibrations of the profile, it is
acted on by loads that are the same as those created if the flow pattern formed
at a given instant of time were to be time-invariant. The use of the "steady-
state hypothesis" for calculating rotor flutter yields quite satisfactory re-
sults which are in good agreement with experiments. Therefore, our entire ac-
count will be based on the results obtained under application of the "steady-
state hypothesis". Refinements that can be made by taking unsteady flow into
consideration will not be examined here.
The use of the "steady-state hypothesis" leads to the following well-known
formulas [see for example (Ref.29, 32, 33)] for determining aerodynamic loads
acting on a vibrating profile of unit length:
r-ic;,^u^[.-iy^{i-f)-^,];]
w„„r=±ebu^l^-f^^_ci^^x
'a.r- 2
xf-i^+(f-f)f»]).
384
(1.5)
■where
T = aerodynamic force per unit length acting on the vibrating profile
In a direction perpendicular to the relative flow velocity U;
^e^er ~ torsional moment per unit length of aerodynamic forces acting rela-
tive to the axis passing at a distance Xg from the profile leading
edge;
9 = angle of blade profile setting in the examined sections;
y = rate of displacement of the blade elements in the flapping plane;
xo = distance between profile leading edge and flexural axis, i.e., up
to the point where the elements of the blade start twisting under
application of a torque;
Of = distance between profile focus or a.c. and flexural axis of the
blade; in some formulas below [see eqs.(2.13) and (5.2)] we will
also use the designation Of-
R
Equations (1.5) sor<3 obtained for a plane-parallel flow. Therefore, their
use for determining the helicopter blade loading is approximate also in this
sense, since the flow past the blade markedly differs from plane-parallel.
It is convenient to make a slight transformation of eqs.(l.5) when using /357
them for the helicopter blade, by Introducing certain additional simplifications
and refinements. The relative velocity U of the flow past the profile can be
approximately equated to its component U^ parallel to the plane of rotation of
the rotor. It must also be considered that the other component of this velocity
Uy directed perpendicular to the velocity U^ differs from y by the amount of the
velocity of the air stream flowing through the rotor. Therefore, for a hell-
copter blade, these formulas are generally
used in the following form:
<"/
—
/
0.3
J
/
-
-
0.2
1
-^
01
T=^\ clQb ]^^ul + U,U,^j--f)bU,<f^ ;
m„
16
Qb^U^^-
o,T.
(1.6)
0.1 0,? 0.3 0.1 as as o.i as 0.9 1.0 m
Fig. 4.3 Position of the Aero-
dynamic Center on the Mach
Number, for a NACA 230 Profile.
The last term in the first equation in
the systems (1.5) and (I.6) ordinarily has
little Influence on the calculation results.
Therefore, it can be neglected without re-
sulting in substantial errors.
In calculations of flutter under condi-
tions of axial flow past the rotor in hover-
ing flight or in operation of the rotor under ground conditions, the aerodynamic
loads can be determined on the basis of the linear dependence of the aerodynamic
coefficient on the angle of attack. This assumption is also included in
eqs.(l.5) and (.16). However, under conditions of forward flight, especially in
regimes close to stalling, this assumption becomes quite inaccurate. Therefore,
a method permitting rejection of this assumption will be discussed below in Sec-
tion 7. Refined formulas for calculating aerodynamic loads for this case will
also be derived in the same Section.
385
As is knovm, the Mach n\imber M has a strong influence on the aerodynamic
characteristics of a profile. To calculate flutter of a helicopter rotor it is
especially important that M have a substantial effect on the position of the
profile focus which, as will be shown below, greatly affects the critical revo-
lutions of flutter. Therefore, in calculations for each blade radius, we must
take the position of the aerodynamic center corresponding to the local value of
M at this radius. Figure 4.3 gives the position of the a.c. as a function of the
Mach number, for a NACA 230 profile.
When calculating flutter in forward flight it must be taken into account
that the local Mach mmiber varies relative to the rotor azimuth. This, in turn,
leads to fluctuations of the position of the profile focus during each revolu-
tion of the blade. In approximate calculations, this circumstance can be disre-
garded .
When using the calculation method presented in Section 7, fluctuations of
the a.c. relative to azimuth can be accounted for without difficulty, which is
one of the important advantages of this method.
Section 2. Flapping . Flutt er_of_ari Is olated Blade with
Axial Flow past tjhe Rotor
7358
1. Blade Model
Flapping hinge axis
Feathering hinge axis
The parameters of flapping flutter can be determined with sufficient reli-
ability from a calculation based on the following assumptions:
l) The blade is absolutely rigid in bending and vibrates in the flapping
plane like a solid body as a
consequence of turning about the
flapping hinge.
2) The blade is absolutely
rigid also in torsion and exe-
cutes torsional vibrations,
rotating like a solid body about
the feathering hinge of the hub
as a consequence of deformation
of the control, presence of an
automatic pitch control mecha-
nism, and installation of a flap-
ping compensator.
Fig.4«4 Blade Model Used in the These assumptions lead to
Calculation. the possibility of calculating
a blade model with two degrees
of freedom, determined by the
variables 3 and cp (Fig. 4.4) • This model is usually called "semirigid".
386
2. Der ivat ion of Differential Equations of Flutter
In this Subsection we will derive the differential equations of flutter for
a model of an isolated blade. It will be shown below, in Section 4, that in many-
cases the theoretical investigations of flutter of a rotor as a whole can be re-
duced to an examination of the flutter of an isolated blade. Therefore, it is
expedient to evaluate first the effect of various factors on the flutter of an
isolated blade and to determine later (in Sect. 4) in what manner and in what
cases these results can be extrapolated to a rotor as a whole.
Let us construct the differential equations of torsional- flapping vibrations
of an isolated blade. These equations can be derived by equating to zero the
sum of the moment of all forces acting on the blade during its vibrations rela-
tive to the flapping and feathering hinges of the hub. As usual, we will examine
small vibrations for which all terms of the second order relative to small dis-
placements of the blade can be neglected.
To avoid needless complication of the equations, let us assume that the
distance from the axis of rotation to the flapping hinge is equal to zero (tq =
= O). Then, the condition of equilibrium of the moments of all forces relative
to the flapping hinge can be written as
R R R
f m (rp - o^) rdr + o)2 J m (r? — ocp) rdr=^T rdr, (2.1)
where /359
B = angle of rotation of the blade relative to the flapping hinge;
cp = angle of rotation of the blade relative to the feathering hinge;
m = mass per unit length of the blade element;
a = distance from the axis of the feathering hinge to the center of
gravity of the blade element ;
T = aerodynamic load per unit length determined by eq.(l.6).
The integrals entering the left-hand side of eq.(2,l) can be expressed in
terms of the moments of inertia of the blade relative to the horizontal hinge
Ih.h and the centrifugal moment of inertia of the blade I^f :
Icf =J mradr.
\
(2.2)
On introducing these designations into eq.(2.l) and referring all terms of
this equation to Ih,h> 'the expression can be rewritten in the form
p + (o2p--^(9- + co2cp) = -J-f rrrfr. (2.3)
'h.h 'h.h J
For the regime of axial flow past the rotor, the velocities entering
387
eq.(l.6) can be equated to
=wr, 1
=u)^X-r^, J (2.4)
where X is the relative rate of flow through the rotor.
Substituting eqs.(2.4) into eq.(l.6), and then eq.(l.6) into eq.(2.3), we
obtain
1
where Yo is the mass characteristic of the rigid blade [see eq.(2.14)].
The values of the coefficients his, ^ig, d^^, and d^g will be given below
[see eq.(2.14)].
The moment of external forces, relative to the feathering hinge, loading
the system that controls the angle of blade setting, can be written as
^cc. = -(9 + '«'<P) fa., +(fi + «>2p)4_^ -M^r-h
«. ^, (2.6)
-^
where
la.i, = moment of inertia of the blade relative to the feathering or axial
hinge ;
In = moment of inertia of the blade per unit length relative to this
axis;
^aer ^ moment of aerodynamic forces per unit length relative to the axial
hinge with this moment being determined by eqs.(l.6):
Mfr = moment due to friction forces in the axial hinge of the hub.
The moment acting on the control system, Meon ^^^ ^^ expressed in terms /360
of rigidity or stiffness and deformation of the control system:
Mc„„=Cc,„\, (2*7)
where
Y = angle of rotation of the blade relative to the feathering hinge due
to deformations of the control system;
Coon ^ stiffness of the control system.
In order to express the value of y i^ terms of the setting angle of the
blade sections, we put
(p = 9-xp + Y, (2.8)
388
where
9 = angle of setting of the blade sections prescribed by the control
system;
H = flapping compensator.
The angle 9 is determined from the expression
9 = 9o— SiSinij)— ejcosij), (2.9)
where
9(3 = angle of blade setting at the root for p = 0;
9i and Q^ = angles of cyclic pitch control.
It follows from eq.(2.8) that
Y=o4-xfi — e.
(2.10)
Substituting y into eq.(2.7) and then eq.(2.7) into eq.(2.6) and referring
all terms of eq.(2.6) to the moment of inertia of the blade relative to the
axial hinge Ia.h> ^^ obtain
Mt
^+(/k,+'"')? + ^2.(P + '«=?) +^p' P+7^-
'■'^ ' ^ch
-it J^«'-'^^=i«+"^^ i J ^".^V^^^-
(2.11)
Here pt« is the frequency of natural vibrations in twist or torsion of an abso-
lutely rigid blade in compliant control:
Pt.-
VI
ypr
a.tr
(2.12)
Substituting into eq.(2.1l) the value of ffiaor from eq.(l.6) and taking the
resultant equation together with eq.(2.5), we obtain a system of differential
equations of binary vibrations of a rigid blade:
f -\r d22^<f + (y^^ + *22"'^) ? + ^2l'^ + ^2l'0f< +
'0 J
rdr.
(2.13)
389
The coefficients entering eqs.(2.13) can be determined by the following /36l
formulas :
V,2 = — - — \mradr=ioC2i,
Co]
== i mrodr,
Ia.h J
2 y /,.;, J V 4 b) '
R
d,A= ^ cj -f-[ br'^o^ dr,
^ 'a.h •)
"22 —
X
_1_
R
*12 =
R
yT-\ brHr-\-ioC^^,
■'A.A J
1
*22=i+Y^;
'a.h J ^
Yo =
c;Qbo.7R^
21
(2.U)
The coefficients of eq.(2.14) entering the differential equation completely
determine the behavior of the blade in vibration. Certain comments are neces-
sary relative to these coefficients.
The damping factor d^^^^ of flapping vibration of the blade is determined
only by aerodynamic forces since the moment of friction forces in the flapping
hinge is relatively small. A quite substantial addition d,, due to friction in
the feathering hinge enters the damping coefficient of the torsional vibrations
of the blade dgg, in addition to aerodynamic damping. The effect of friction in
the feathering hinge -will be discussed in greater detail in Section 3 of this
Chapter .
The coefficient d-^g entering the equation is small and not essential for
390
the final results of the calculation. Therefore, it can be disregarded in
practical calculations.
If the ratio of the moments of inertia io
-a. h
-, then we can /362
Ih.h 1000
also neglect the coefficient c^g. In so doing, the system of equations (2.13)
is simplified even more.
It is important to note that the effect of the position of the center of
gravity of the blade element will appear in the calculation only upon a change
in the coefficient
1 mra dr.
(2.15)
3. Particular Solution_ of the^ Diff erential Equation
It is not difficult to demonstrate that the expressions
fj* = flo — a, cos <J» — 6j sinij;,
9* = ?o — <Pi cos tl* —'^i sin i>
(2.16)
are a particular solution of the system of differential equations (2.13) and de-
termine the undisturbed motion of the blade. If the swashplate of the automatic
pitch control is set in a neutral position and if 9i = Qs = 0, then the particu-
lar solution of these equations is constituted by the expressions
9* = ?o-
(2.17)
4. Differential Equation o f Disturbed Motion
Let us substitute into eq.(2.13)
P = P* + Prf,
9 = 9* + ?^,
(2.18)
where B^ and cp^ are the angles of deflection of the blade from a position cor-
responding to its undisturbed motion.
Then, bearing in mind that 3* and cp* represent the particular solution of
eqs.(2.13), we obtain a system of differential equations of disturbed motion of
the blade:
9 + rf22">=<P + (/^^ +*22'"=) ? + C2,F+ d,,4^-(:'-Pl +^2l">=) P =
Ac
(2.19)
391
■•0
In these equations, the subscript of the variables 3 and cp, designating
that these vari^les refer only to disturbed motion, is dropped for simplicity.
5. Notation of Differential Equations in Matrix Form
It is convenient to vrrite differential equation (2.19) in the following
matrix form:
CX-\-D,>iX-\-{A-\-m'iB)X=Q. (2.20)
Here, C is the inertia matrix:
D is the damping matrix: /363
l<., tin J
A is the stiffness matrix;
B is the matrix of centrifugal and aerodynamic forces;
where bgi = Cg^ .
X is the vector function:
6. Solution of Differential Equations of Blade Vibrations
Setting, in the system of equations (2.19),
(2.21)
we obtain the following characteristic equation:
X< + A^l^+ (5,^24-52)'X2+(Ci«HC2)wX+D,»*4-D2;;^2=0. ( 2.22)
392
Here for simplifying the calculations, the values of X and cu are referred
to the frequency of natural torsional vibrations of the blade pt«, i.e..
X = -
''fu
^/■w
(2.23)
The coefficients entering the characteristic equation (2.22) have the
following form:
A,.
. "2~(''n~l~^22 — <^12''21 — ^2l"l2)i
. ~(1 4" ^22 ~r ^11*^22 — ^I2<^21 — "12"21 — ^12^2l)l
— (1-xr,.^),
VJi
. "2~ ('^22"r'^Il''22 — ''l2"21 — <'2l"l2)>
0^21
(cfii — xoTij),
'0621
— (^22~^12'^2l''
'0''2I
(2.24)
Let us examine the behavior of the roots of the characteristic equation /364
(2.22) for different rotor parameters.
In the major portion of the rpm range of practical interest, the motion of
the blade is determined by two pairs of roots:
^/ =gi±ipi'i/ic/'^// = q2~fiP2-
(2.25)
Figures 4.5 and 4.6 show the dependence of the real and imaginary parts of
these roots on the rotor rpm and on the blade balancing. In both graphs, we
plotted, on the abscissa, the rotor rpm n related to the frequency of natural
vibrations of the blade in torsion pt^, expressed in oscillations per minute:
The values of n coincide in magnitude with the values of the relative
angular velocity
393
p,.pi
to
as
as
0.1
0.2
C„ ''0.8(?3.5''/o)
"Iff (21%) /
- - — - /■
C,,='3.0(*£S%)
He I icop ter
rpn range
Jf
A,
m
/4
^
/y
//
N
k
/
/
r
•\
^"\
Pi
M.
./
y
-0(23%)
C2,'=0.8{23.5%)
\
N
\
/-
=!.6(i
'I'M
\
\
\
^
\
\
0.2
01
0.6
Fig.4.5 Imaginary Part of the Roots
of the Characteristic Equation as a
Function of Angular Velocity, for
Different Values of the Coefficient
Cgi . [In this diagram, as well as
in Fig.4«6, we indicate the absolute
value of Cgi (without the minus
sign).]
where pt„ is expressed in rad/sec.
Therefore, we will henceforth use
the designations n and cju on an equal
footing.
The roots of the characteristic
equation determine the law governing
the motion of the blade after some
extraneous action (in practice, this
may be - for example - a gust of wind)
unbalances the blade. In this case,
the value of the real part of the root
q determines the rate at -which the
amplitude of the vibrations varies,
whereas the imaginary part p determines
their frequency. The negative real
part of the root corresponds to damping
oscillations of the blade. When this
quantity is positive, vibrations of an
amplitude increasing in time will be
generated.
The first pair Xj , shown in
Figs. 4. 5 and 4*6 by broken curves, de-
termines the motion in which deflection
of the blade relative to the flapping
hinge is ^predominant. The second pair
of roots Aj J , shown by solid curves,
determines the motion with an appreci-
able rotation of the blade relative to
the feathering hinge which is due to
deformation of the controls.
This second motion is of greatest
interest since, at certain blade balancing, the real part of the root qg passes
into the area of positive values (see Fig. 4. 6), which corresponds to vibrations
of increasing amplitude, which are known as flutter. /365
The values of the rotor rpm at which qg = are usually called "critical
rpm of flutter".
When qg < 0, the blade executes damping oscillations. In this case, the
value of qs determines the magnitude of forces that produce damping of the blade
vibrations and constitutes a criterion for their stability. It follows from
Fig. 4. 6 that the damping forces begin to decrease long before the critical
flutter rpm. This decrease is observed even when flutter cannot arise no matter
what the rotor rpm but the margin for blade balance is insufficiently narrow.
A decrease in aerodynamic damping, and hence of stability of blade vibrations,
is undesirable and may have an adverse effect on the characteristics of heli-
copter controllability.
394
-0.Z
-0.B
-f.O
Fig. 4.6 Real Part of the Roots of the Characteristic
Equation as a Function of Angular Velocity, for
Different Values of the Coefficient csi .
The peculiarities of the behavior of the first pair of roots \ will be
examined below in Subsection 8.
7. Determination of the Critical Flutter Rpm
To determine the critical flutter rpm, it is possible to derive an ana-
lytical expression if, in the characteristic equation, we set
X==tyr,2=i^^^; (92 = 0)
(2.26)
Then, the characteristic equation (2.22) reduces to a biquadratic equation
where
2L =
C, {2C2 — .4,62) + Ax (yli Os — B1C2)
7366
(2.27)
(2.28)
395
k
M--
Cj(C2-^,B2)
from which we can determine the critical flutter rpm
^flu=VL^Vl^^^^- (2.29)
The vibration frequency in flutter is determined from the expression
^,.-l/^^^- (2-30)
8. Blade Divergence
A study of the graphs in Fig.4.6- indicates the behavior of the first pair
of roots Xj .
Beginning with a certain rotor rpm, the imaginary part of this pair vanishes
and two real roots appear. The presence of real roots indicates aperiodic mo-
tion of the blade.
With a further increase in rpm, one of these roots A.j ^^ passes into the
region of positive values, which characterizes the appearance of aperiodic in-
stability at this rpm, known as blade divergence.
The value of the rotor rpm at which X. = is known as the "critical rpn
of divergence" and can be determined by the formula
Usually, the critical divergence rpn is higher than the critical flutter
rpm and the maximum rotor rpn. However, in a number of special cases, blade
divergence is a decisive factor. For example, the possibility of the occurrence
of divergence does not permit using negative values for the flapping compensator.
At H = 0, the possibility of occurrence of divergence is already quite real, and
at small negative values of k the blade becomes aperiodically unstable. This
circumstance must be taken into account when designing the rotor hub, especially
when deflection of the blade relative to the drag hinge kinematically leads to
a decrease in the values of k to below zero.
9. Parameters Characterizing Blade Bala nce (Ef fective
Blade Balance) "
To evaluate a blade from the point of view of possible flutter, it is con-
venient to introduce several concepts characterizing the position of the e.g.
of blade elements over the blade length. The quantity
396
==f (2.32)
is called blade balance in a given section.
If balancing of the sections is constant over the blade length, then the /367
value of the coefficient c^^ entering the equations will be directly related
with the magnitude of this balance. The flutter characteristics of a blade in
this case can be characterized by the value of the balance of its sections.
In practice, however, balancing of blade sections lengthwise is always dif-
ferent. Therefore, it is convenient to evaluate its flutter characteristics by
means of the so-called effective balancing.
The effective balancing of the blade in question is defined as the balanc-
ing of some equivalent blade with an identical rotation of the centers of gravi-
ty over the length and having the same value of the coefficient Cgi . It is con-
venient to .assume the planform and mass distribution over the length of the
equivalent blade as being identical to those of the blade in question. In this
case, the effective balancing of the examined blade can be determined by the
expression
0»»r =
R
J mar dr
— Cixla.h
<//~" R — R
f mbr dr J" mbr dr
(2.33)
For blades having the axis of the feathering hinge at a distance constant
_ Xp
in percent of the chord from the leading edge Xq = -^ — = const, it is convenient
to characterize the effective balancing of the blade by the value of balancing
of an equivalent blade relative to its leading edge
-««//= -^0 +"«//• (2.34)
Since the position of the axis of the feathering hinge has only a slight
effect on the values of the critical flutter rpm, it is convenient to reckon
effective balancing from the leading edge also in cases in which the condition
-T- — = const is not satisfied. Then, the effective balancing can be determined
by the expression
R
J mar dr
^y/=y-— • (2.35)
J mbr dr
The effective balancing of manufactured blades can be determined only by
397
cutting the blade and experimentally determining the balancing of its individual
segment s .
10. Dependence of Critical Flutter Rpm on Blade Balancing
and Values of the Flapping Compensator Coefficient
To illustrate the effect of various parameters on the critical flutter rpn.
Figs. 4.7, 4.8, and 4.9 give the results of calculations performed by eq.(2.29).
The curves refer to different values of the flapping compensator coefficient k
and to three values of the position of the feathering hinge axis xq at a constant
position of the profile focus.
The graph shows that a shift of the e.g. toward the leading edge, just as a
decrease in the flapping compensator, will improve the flutter characteristics
of the blade, whereas a shift of the e.g. toward the trailing edge and an in-
crease in the flapping compensator will lead to a decrease in the critical
flutter rpm. These results coincide qualitatively with experimental data.
7368
Fig.4.7 Critical Flutter and Divergence Fig. 4. 8 Critical Flutter and Diver-
Rpm as a Function of Effective Blade gence Rpn as a Function of Effective
Balancing, for Xq = 0.18. Blade Balancing, for Xo = 0.23.
A comparison of the results of calculations performed for three different /369
positions of the feathering hinge axis shows that the effect of this parameter
on the critical flutter rpm is incomparably weaker than the effect of blade
balancing. Consequently, the critical flutter rpn depends mainly on the mutual
398
za'/o
2'^%
25%
26%
^Vf
Fig. 4. 9 Critical Flutter and Di-
vergence Rpm as a Function of Ef-
fective Blade Balancing, for
xo = 0.28.
position of the centers of gravity of
the blade elements and of the profile
focus. Therefore, a shift of the a.c.
of the profile relative to the chord is
just as effective as a shift of the
blade balance.
11. Blade_ Arrangement
The presented dependences of the
critical rpm on the balancing permit
necessary" conclusions with respect to
blade arrangement. It follows from the
above calculations that the best way to
improve the flutter characteristics of
a blade is to shift its centers of
gravity as much as possible toward the
leading edge and to use aerodynamic pro-
files which, in operating flight regimes,
have their aerodynamic centers as far
rearward as possible. This measure has
a favorable effect even when the blade
spar is shifted toward the leading edge
to create forward balance, together with
the feathering hinge axis which often is
associated with the axis of the spar.
The arrangement of the blade shown in
Fig. 4 .10 is an example of such a solution.
However, it must be borne in mind that the statement as to the relatively /370
weak influence of the position of the feathering hinge axis on the flutter
characteristics in comparison with blade balancing holds true only when the
variation in these parameters is
of the same order of magnitude.
In practice, a shift in the posi-
tion of the feathering hinge can
be performed in appreciably wider
limits than a shift in blade
balancing. Therefore, this should
be regarded as still another means
of influencing the blade flutter
characteristics.
The blade whose arrangement
is shown in Fig. 4.11 can serve as
an example for the case in which
a change of the position of the
Fig. 4. 10 Blade Arrangement with Feather- feathering hinge is used as a
ing Hinge Axis and Spar Shifted toward means of improving the flutter
the Leading Edge. characteristics.
399
rm
12. Effect of Control Rigidity
A highly important parameter
greatly influencing the flutter
speed is the magnitude of the fre-
quency of natural blade vibration
in torsion or twist ptw . In the
idealized blade scheme examined
here, the magnitude of this fre-
quency is completely determined by
the hinge rigidity of the system
controlling the angle of rotor
setting Coon» I'^ practice, how-
ever, the magnitude of this fre-
quency is influenced also by tor-
sional deformations of the blade
itself. Therefore, to take into
account the torsional rigidity of
the blade it is proposed to use, in calculations by the approximate method pro-
posed here, the value p^^ calculated with regard to deformation of both the con-
trols and the blade.
Fig. 4. 11 Arrangement of Blade with
Turned Feathering Hinge Axis.
We see from the differential
the critical flutter rpm (flutter
'«//
26%
P«%
22%
^one of possible
flutter at low
control rigidity
equations of blade vibrations [eq.(2.19)] that
speed) and frequency of vibrations in flutter
are directly proportional to the
quantity Pt„. Therefore, in all calcu-
lations whose results are presented in
the above graphs, the flutter speed is
referred to p^^ and is characterized
by the relative quantities
Zone of impossibility
of flutter
OM
0.8
12 X
Fig.4«12 Boundaries between Zones
in which Flutter is Impossible and
the Zone in which it Arises at
Small Control Rigidity,
o - Rotor blades for which no
flutter was observed
n - Rotor blades for which
there was flutter.
- n.iu
(2.36)
13. Conditions for Absence of Flutter
The character of the dependence of
flutter speed on various parameters
shows that the creation of the neces-
sary flutter characteristics does not
require a simultaneous change of all
parameters. Production of the necessary
characteristics is possible upon satis-
fying even one of the two following
conditions:
The first condition is the
creation of a sufficiently high tor-
sional rigidity of the blade and its
attachment to the control system, so
that
/371
400
p. >Mn
(2.37)
Here, rinax is the maxlm.'um possible rotor rpm. It is sufficient that kx =
= 4-5.
When the condition (2.3?) is satisfied, there is no need to secvire any spe-
cific transverse blade balancing. It can be arbitrary, and there is no need for
introduction of special counterweights into the design.
The second condition is the creation of a sufficiently forward blade balanc-
ing so that
V/<-^"
h'lf
(2.38)
Here, Xna is some limiting blade balancing at which flutter is impossible
no matter how small, say, the torsional rigidity of the blade attachment to the
control.
Figure 4«12 gives the calculated value of the limit balancing Xii„- as a
function of the value of the flapping compensator and position of the feathering
hinge axis 5Ea. This balancing divides the entire area of parameters into two
zones, in one of which flutter cannot occur even at very low control rigidity.
The available statistics on full-scale flutter tests of rotors with blades
whose effective balancing after the test was determined by cutting, satisfacto-
rily agree with the described limit ( see
Fig. 4. 12).
Experience has shown that, in
practice, only one of the indicated con-
ditions is more readily met.
14. Mechanism of Generation of Forces
Exciting Flutter
Fig. 4 .13 Damping Forces Acting on
a Vibrating Profile without Tor-
sional Vibrations of the Blade.
ing of this phenomenon.
The calculation methods that reduce
to a determination of flutter parameters
are^ left without an explanation of the
mechanism of action of aerodynamic forces
leading to the generation of divergent
vibrations. A detailed study of the
nature of the aerodynamic forces acting
in flutter yields no new data for flutter
investigations. However, in certain
cases it does promote better understand-
Let us examine the blade model which was described in Subsection 1 of this
Section. For simplification of the problem, we will limit ourselves to the
401
particular case vjhere the aerodynamic center coincides vrith the axis of the
feathering hinge and -hAiere Of = 0. We can also disregard the dependence of the
force T on cp, which does not have any particular meaning. Then, the aerody- /372
namic forces acting on the profile can be represented in the form
T = ±-c'^QbU^a; (2.39)
2 y
Te
where a is the angle of attack of the blade element.
^..r=-^Qb'Ui (2.40)
The moment of the aerodynamic forces T acting relative to the flapping
hinge can be written as
^*.A=*°' (2.41)
where
/?
'- = ~c;Q<.^jbr^dr. (2.42)
We will assiime that the blade executes vibrations relative to the flapping
hinge according to the law
? = PoSinp(. (2.43)
In this notation, the time reference point is taken from the instant at
which P = 0.
First, we will examine the case in which the blade does not execute tor-
sional vibrations. The angle of setting of its elements will be considered as
equal to zero and constant in time. In this case, the angle of attack of the
blade elements will vary according to the law (Fig. 4.13)
a=acos pi, (2-44)
where
"=-^»f- (2.45)
The moment of the aerodynamic forces relative to the flapping hinge will
vary by the same law
M^j,=M cos pi. (2.46)
In accordance with eq.(2.4l), the sign of M will coincide with the sign
of a.
If a < 0, as occurs in the case in question, then the moment relative to
the flapping hinge always acts opposite to the angular velocity of blade vibra-
tions P (see Fig. 4. 13) and does negative work in blade displacements.
402
The magnitude of this work diiring the vibration period can be calculated
by the formula
2it
p_
A = ^M^I^^dt=.^M%pQ.o%'^ptdt=n%M, (2.47)
Ott
where T = ■ is the period of blade vibration.
P
The sign of the work A coincides with the sign of M which, in turn, coin- /373
cides with the sign of a. In the examined case, A < 0.
This means that the air stream flowing past the blade absorbs the work ex-
pended to maintain blade vibrations. Thus, in the presence of aerodynamic forces
the blade will vibrate with a constant amplitude Pq only if energy equal to the
magnitude of work calculated by eq.(2.47) is furnished to it from without.
Otherwise the kinetic energy of the blade and, together with it, the amplitude
of oscillations Po > will diminish and the oscillations will decay.
A different picture may be produced in the presence of torsional blade vi-
brations. Torsional vibrations of the blade arise as a consequence of deforma-
tions of the control system and kinematic coupling across the flapping compen-
sator. Deformations of the control system arise from aerodynamic and inertia
forces acting on the blade during its flapping vibrations.
Centrifugal and inertia forces arising during .flapping vibrations of the
blade create a moment relative to the feathering hinge due to the presence of
an arm between the centers of gravity of the blade element and this axis
'nmtrt= ~{p^ — w'^)?o^ mar dr Sin pt. (2.48)
The aerodynamic forces create a moment on the arm between the profile focus
and the feathering hinge axis Of
R
fna<ir= clQuip'pA br^oj, drcospi. (2.49)
At Of = 0, this moment is equal to zero. Therefore, as a consequence of
flapping vibrations only the moment mj„ej.t will act on the blade. Under the
effect of this moment, the blade pitch control is deformed and the blade begins
to execute torsional vibrations. However, the phase of the torsional vibrations
will not coincide with the phase of the flapping vibrations. Phase shift of the
torsional vibrations is caused by damping forces acting in the control system
directed opposite to the vibrations. These forces are caused by forces of aero-
dynamic damping determined by eq.(2.40) and by the moment of friction acting in
the feathering hinge of the blade. The direction of phase shift of the torsional
vibrations depends on the sign of the external moment mj^gyt*
The law according to which the blade executes torsional vibrations
403
(Fig. 4. 14) can be written as
tp = 9 cos pt -\-os'm pt.
(2.50)
Here, it is assumed that the initial setting of the blade elements is equal
to zero.
The angle of attack in this case will vary according to the law
a =a cos/?/ -fa sin /7^, (2.51)
where
«=?-8o— ;
a-=f.
The appearance of a sinusoidal component in the law of change of the angle
of attack a and, along with this, the sinusoidal component of the moment relative
to the flapping hinge, does not influence the energy transfer during blade /374
vibrations. Actually, a check on the
work done by the sinusoidal component
of the moment H in blade displacements
relative to the horizontal hinge will
show that it is equal to zero:
A^^^Msm pt p^Q cos pt dt -
=0.
(2.52)
Fig. 4.14 Damping Forces Acting on
a Vibrating Profile in the Presence
of Torsional Vibrations of the
Blade .
The magnitude of the co sinusoidal
component of the angle of attack a, as
follows from eq.(2.5l), largely depends
on the sign and magnitude of cp.
When 9 < 0, the work absorbed by
the air stream flowing past the blade
increases which causes a rise in the
rate of damping of the free vibrations
of the blade. Thus, when cp < 0, the
stability of flapping vibrations of the
blade increases. When 9 > 0, the work
absorbed by the stream past the blade
decreases and vihen
^=^0^- (2.53)
it becomes equal to zero, whereas when
^>^°f- (2.54)
the cosinusoidal component of the horizontal hinge moment is directed along the
404
angular velocity of the flapping vibrations 3 . This leads to "resonant build-up"
of the blade. The kinetic energy of blade vibrations begins to increase, which
leads to a rise in the vibration amplitude. Such a type of oscillation at
amplitude build-up is known as flutter.
Thus, the occurrence of flutter is associated with the magnitude and sign
of the component of torsional vibrations cp.
Let us examine how the quantity cp changes under the effect of an external
moment varying by the sige law in conformity with eq.(2.48). Figure 4.15 shows
the dependence of cp and cp on the vibration frequency p of the external moment
miner t. As usual during vibrations close to resonance, the component 9 which is
in 90° phase with the external forces first increases, whereas the vibration
component coinciding in phase with the external forces changes its sign in
resonance, passing through zero.
Thus, the value of cp increases especially upon approaching resonance with
the frequency of natural blade vibration in torsion. Therefore, flutter always
occurs with a frequency close to but slightly below the frequency of torsion. /375
Usually the frequency of flutter amounts to about 0.8 Ptw*
It follows from the foregoing that flutter occurs as a consequence of the
following causes: The torsional moment
due to inertia forces acting during
V',f° 7 11 I ( I l/TM I I \ \ 1 — I flapping vibrations of the blade leads
to the appearance of torsional blade
vibrations. In so doing, the torsional
vibrations with a 90° phase shift rela-
tive to the flapping vibrations increase
especially strongly at frequencies
close to the frequency of the natural
vibrations of the blade in torsion.
This component of the torsional vibra-
tions leads to excitation of flapping
vibrations of the blade. As soon as
this excitation [first term in
eq.(2.55)] becomes stronger than the
forces damping the flapping vibrations
[second term in eq.(2.55)], flutter
will occur.
-2
-3
-U
-
'S
i—
^
i~
r^
V
\
-
\
9-
i
^
\
-
-
-
)
r--
—
~~
/
\
/
\
/
/
\,
/
\
S
~^
/
/
1
s
i
1
^
y
I
1
"^
zoo
100
eoo
P
s.
>
\
1
/I
;
1
—,
p
\
k
t
.y
Fig. 4. 15 Variation in the Torsional
Vibration Components cp and cp during
Blade Vibration Frequency.
From the expression for the co-
sinusoidal component of the angle of
attack
5=»-Po-^ (2.55)
it is also possible to trace the effect
of rotor rpm on flutter. Actually, the
second term in this formula rapidly
decreases with increasing rotor rpm,
whereas cp does not greatly depend on
405
the rpE since the external torsional moment xa^j^ert i^ determined mainly by the
vibration frequency [see eq.(2.48)] because of the fact that, during flutter, p^
usually is by a factor of 5 - 8 greater than cu^. The variation in 9 -with respect
to rotor rpm is related mainly -with an increase in aerodynamic damping at increas-
ing ou.
Thus, on tracing the mode of variation of the quantities entering eq.(2.55)
with the rotor rpm, it will be found that, at some value of w, the cosinusoidal
component of the angle of attack a changes in sign and becomes positive. This /376
leads to the appearance of flutter, beginning with some specified rotor rpm. A
rearward shift of blade balancing leads to an increase in the absolute value of
m.
Inert
[eq.(2.49)] and hence to an increase on 9. In this case, as follows from
eq.(2.55), flutter arises at smaller
u>.
In the same manner, it is possible to trace the effect of various other
parameters on the flutter speed. However, there is no need for this since this
has already been done above with sufficient detail.
Section 3. Consideration of Friction Forces during Flutter
1. Character of the^ffect of Friction_ j^orce£ du ring Flutter
The occurrence of flutter leads to the appearance of oscillatory motions in
the hinges of the rotor hub and in the hinge control. Therefore, the friction
forces acting in these hinges have a substantial effect on the critical rpm and
on the nature of generation of flutter.
Of primary importance in this case is
friction in the feathering hinge of the
blade loaded by a centrifugal force, in
comparison with which the friction in
all other hinges can be neglected.
Experiments show that forces act-
Fig. 4. I6 Recording of the Moment ing in the feathering hinge are similar
of Friction in the Feathering in character to forces of dry Coulomb
Hinge during Torsional Blade friction [eq.(4.l6)]. The introduction
Vibrations. of these forces into the calculation
makes the problem of flutter essential-
ly nonlinear. Therefore, in simplified
calculations it is natural to use any of the possible methods of linearization
of friction forces. A more exact solution to this problem without such lineari-
zation will be given in Section 7 of this Chapter.
As is known, linearization of friction forces leads to the dependence of
the damping coefficient on the amplitude of oscillations. Here the nature of
flutter generation, described on the basis of the calculation changes at in-
creasing amplitude, approaching that observed in experiments on helicopters.
These results permit explaining numerous peculiarities in the development of
flutter in full-scale experiments. The possibility of interpreting these char-
acteristics appreciably facilitates the conduction of tests.
406
2. Lin earization of Friction Forces
Let us use the energy method of linearization of friction forces. For this,
we will replace the moment of friction acting in the feathering hinge of the
blade by some equivalent moment whose magnitude is proportional to the rate of
angular blade displacement
^f.^=-Y,.9. (3.1)
The value of the coefficient Yfr is determined from the condition of equali-
ty of the work done during the vibration period by the moment of friction,
A^r=4M^rVfio (3.2)
and by an equivalent moment whose magnitude is proportional to the vibration /377
rate
^^?="V^//^/"' (3.3)
where
Mfr = constant (in magnitude) moment of friction acting in the feathering
hinge, always opposite to the 'rate of relative displacement;
cpfiu = amplitude of torsional blade vibrations in the feathering hinge
during flutter;
Pfiu = frequency of blade vibrations during flutter.
The moment of friction acting in the feathering hinge can be considered
proportional to cu^, since its magnitude is determined mainly by the centrifugal
force
In a number of cases, however, this dependence is disturbed as a conse-
quence of the following circumstances:
1) The bearing is Installed with appreciable prestressing. In this case,
the load acting on the bearing is determined not only by centrifugal force but
also by the Initial tension.
2) The design of the packing glands is such that they have an appreciable
moment of friction regardless of the magnitude of the effective centrifugal force.
3) The use of too heavy a lubricant in the bearing creates an appreciable
additional moment due to the generation of viscous friction. The appearance of
relatively large viscous friction forces is often observed at low negative tem-
peratures of the ambient air.
All these facts have an influence on the flutter speed but introduce no
fundamental features into the pattern of the phenomenon. Therefore, in the
following account we will take eq.(3.4) as the basis.
The coefficient af^ entering eq.(3.4) is determined from the expression
407
<^f=K^a^^ (3.5)
where
Sa.r - static moment of the blade relative to the axis of rotation;
r^g = radius of the thrust bearing;
f = coefficient of friction in the bearing.
The values of the friction coefficients f are usually quite stable and
amount to about 0.003 for ball and 0.006 for roller bearings.
After equating eqs.(3«2) and (3.3)» we obtain the expression for determining
the coefficient Yf r •
^f'^^ "^lii^ ' (3.6)
With this method of linearization, consideration of the friction forces
leads to only one change in the initial equations (2.19), namely of the coeffi-
cient dss standing for the first derivative of the angle of rotation of the
blade in the hinge, which is supplemented by some addition d, ^ .
In an investigation of flapping flutter with a blade rigid in torsion, this
supplement should be determined by the formula
^F = -r^ • (3.7)
3 . Determination of Flutter Speed with Considerati on of F riction /378
Equation (3.7) derived above, which determines the magnitude of the addi-
tion term due to friction forces to one of the coefficients of the equations of
blade vibration d^s, is distinguished by a highly important characteristic.
This addition depends on the amplitude of blade vibration in the feathering
hinge during flutter cpfiu* Consequently, the critical rpm at which the ampli-
tude of oscillations theoretically remains constant in time depend on the ampli-
tude of flutter oscillations.
Figure 4.17 shows such a dependence for three values of blade balancing ob-
tained in a calculation of flapping flutter. Along the abscissa in this diagram
is laid out the amplitude of angular blade vibrations in the feathering hinge
cpfiu, and along the ordinate the critical flutter rpa referred to the frequency
of natural vibrations of the blade in torsion Hj lu .
These curves determine the amplitude of the oscillatory regime, which forms
the boundary between oscillations with amplitude build-up and damping oscilla-
tions.
For all practical purposes, this means that for flutter to occur some
initial impetus is needed leading to deflection of the blade from a position of
equilibrium by an angle determined by these ciicves, usually called the excita-
408
Fig. 4. 17 Critical Flutter Rpm as a
Function of the Vibration Amplitude
9f lu •
quencies.
tion threshold.
If there is no such impetus present,
flutter will not occur at all no matter
what the rotor rpm' might be.
For a comparison. Fig. 4. 17 shows
the critical rpn for the case in which
the moment of friction in the feather-
ing hinge is Mfj. = 0.
4. Effect of Forced Motion in the
Feathering Hinge
Quite a different picture of the
occurrence of flutter is observed when
forced motion is present in the feather-
ing hinge of the hub caused by tilting
of the swashplate of the automatic
pitch control or by forced flapping
vibrations of the blades arising in
flight during oblique flow past the
rotor. In this case, the vibrations
in the feathering hinge following the
occurrence of flutter are generated by
a complex law consisting of two oscil-
latory motions with different fre-
Figure 4«18 shows, as an example, the pattern of this motion observed
during flutter under conditions of ground tests when forced motion is present in
the feathering hinge caused by tilting of the swashplate of the automatic pitch
control (curve cp^ or ) ^^'^ 'the motion caused by flutter (cpfiu). /379
■/?V^^
Oscillation period
Fig. 4. 18 Character of Flutter in the Presence of
Forced Motion in the Feathering Hinge.
For convenience of further discussion, we plotted the rate of vibration in
the feathering hinge rather than the displacements.
The work of the friction forces acting in the feathering hinge can be
409
determined by the expression
t
A
^r=]Mf,kdt, (3^gj
where the moment of friction Mf^ is always directed opposite to the rate of
angular displacement of the blade 9.
If the rate of angular motion cp is the sum of two oscillatory motions
then the work of the friction forces can always be represented as consisting of
two works, in each of these motions
^/' = V+^/''" (3.10)
where ^
t
^flu=^^f9fl„dt.
ti
Here, the moment of friction - as usual - is directed opposite to the rate
of total motion cp.
The simultaneous presence in the feathering hinge of two oscillatory mo-
tions of different frequency always leads to the appearance of time segments
during which the friction force coincides in direction with the rate of one of
these motions, in this case doing positive work. In Fig. 4.18 the area segments
corresponding to the positive work of friction forces in displacement of one of
the composite motions of a frequency Pf lu are hatched. As a result, the overall
magnitude of work of the friction forces during the vibration period in dis-
placement of each of the composite motions decreases in comparison with the case
where there is no concomitant motion. As applied to otir case, this means that
the work expended for damping flutter vibrations markedly drops because an ap-
preciable portion of the friction forces is expended by forced motion. This
drop can be characterized by a special coefficient which represents the ratio /380
J Aff
where
Af lu ~ work of friction forces during the vibration period in displace-
ments of the component of motion caused by flutter, which is of
interest here;
Afr = work of friction forces during the same period when there is no
concomitant forced motion.
Figure 4.19 shows the dependence of the coefficient Afj^ on the amplitude
410
ratio of the velocity components of oscillatory motion ( ~) *
^for _
At the values of ^^'■" = 1.5 - 2.5 of interest to us, the coefficient Afi„
Pf or
depends little on the ratio of these frequencies.
If the value of the coefficient Yfr is determined in this case, as was done
above, from the condition of equality of work [see eq.(3.6)], then eq.(3»7) takes
the following form:
4a/r-^//a (0 1
«4.A. Pfiu yiai
(3.12)
the
It follows from this expression that the critical flutter rpm depending on
coefficient d,, is related with the amplitude of forced motion in the axial
fiflu
—
-'-
—
7
^
1
/
/
/
/
ff^
/
/
/
OM
0.8
1.2
I.S
Fig. 4 .19 Dependence of the
Coefficient Sfj^ on the Ampli-
tude Ratio of the Velocity
Components of Oscillatory
Motion.
hinge cp.
*-nu
since d,, depends on the quantity
With consideration of the nonlinear
dependence Af ^ „
"for
shown in
Fig. 4. 19, this relation becomes rather com-
plex. However, consideration of this de-
pendence radically changes the character of
the conditions necessary for the occurrence
of flutter.
Figure 4-20 gives the values of critical
flutter rpm at different magnitudes of the
oscillatory blade motion in the feathering
hinge cpfor> calculated with consideration of
this nonlinear dependence as applied to flap-
ping flutter. The calculation was made only
for one value of blade balancing and different
amplitudes of forced motion in the feathering
hinge cpf o r •
The curves plotted in Fig. 4. 20 permit a
number of interesting conclusions.
First of all, it follows from these curves that, in the presence of forced
motion in the feathering hinge, flutter occurs at certain revolutions of the .
rotor and its appearance is not due to the effect of any extraneous influence in
the form of some initial impetus. In this case, the rpm of flutter onset is
smaller, the greater the amplitude of forced motion in the feathering hinge cpfor »
This fact is responsible for the dependence of the critical flutter rpn in /381
flight on all parameters of the flight regime that determine the amplitude of
cpfor* 3^<i primarily on the helicopter balancing and the flying speed. In ground
tests, this leads to dependence of the critical rpm on the position of the con-
trol stick.
A second important characteristic of flutter, following from the curves
411
(see Fig. 4. 20), is the appearance of two different types of flutter which differ
by the character of the increase in vibration amplitude upon any change in rotor
rpn.
Upon an increase in rpn to values
flutter win set in with an . amplitude
Fig. 4. 20 Variation in Critical
Flutter Rpm with Vibration Ampli-
tude in the Feathering Hinge cp, j^,
at Different Magnitudes of Forced
Motion.
"hard flutter".
corresponding to the points ai, ag, as,
smoothly increasing with increasing rotor
rpm. If, after the occurrence of
such oscillations, which are usually
called "soft flutter", the rotor
rpm remains unchanged, then their
amplitude will remain constant for
as long as desired.
Oscillations of this type have
been repeatedly observed in ground
and flight studies of flutter in
helicopters. A decrease in rotor
rpn after the occurrence of "soft
flutter" leads to cessation of
oscillations at the same rpm at
which flutter began.
Upon an increase in rotor rpm
to values determined by the points
bi and h^, oscillations are generated
whose amplitude increases in time
without an increase in rotor rpm.
Oscillations of this type are called
Probably, the limiting values of the vibration amplitudes obtainable in
this case are determined by the nonlinear nature of the change in aerodynamic
forces relative to the angle of attack. This branch of the curve in Fig. 4.20 is
shown approximately by a dashed line.
When "hard flutter" occurs during ground tests of a helicopter, the in-
crease in blade vibrations can be stopped (to prevent an accident) only by a
marked decrease in rotor rpm. The generation of such oscillations in flight may
lead to serious consequences.
A decrease in rotor rpn after the onset of "hard flutter" leads to cessa-
tion of vibration at an rpm corresponding to the point k, which, as a rule, is
smaller than the values corresponding to ax and ag .
Thus, to stop "hard flutter" the rotor rpn should be decreased to values
lower than those at which flutter began.
At small amplitudes of forced motion in the axial hinge, the occurrence of
"hard flutter" is possible only after some initial impetus, just as in the case
when forced motion is absent.
The rpm corresponding to the point nx should be considered the most
7382
412
probable rpm for the start of "hard flutter" since, in this case, the magnitude
of the necessary impetus is minimal.
In calculating the critical rpm for the onset of flutter, corresponding to
ai, as, as in Fig. 4. 20, additional simplifications can be made in eq.(3.12).
As follows from Fig.4.19, when — ^ ^" <, 0.5, the value of the coefficient
_ Vf or
Afjy can be determined by the formula:
If the frequency of forced motion is p, 5,, = mu) (m being the order of the
harmonic of this motion with respect to rotor rpm), then we can write
In this case, eq.(3.12) takes the following form:
dfr--- ,'"''' . (3.14)
The value of the equivalent moment of friction is here proportional to the
rate of angular displacements and does not depend on the vibration amplitude of
flutter cpf 1 u :
^^i^-i^a;:^'^- (3.15)
In other words, the moment of friction acting in the feathering hinge in
the presence of forced motion in this hinge affects small oscillations of the
blade in the same manner as a linear vibration damper, whose moment is propor-
tional to the rate of relative displacement. This conclusion pertains not only
to the feathering or axial hinge of the blade but is generally valid for all
mechanisms with friction.
It also follows from Fig. 4*20 that friction in the feathering hinge, even
in the presence of forced motion, increases the critical flutter rpE in compari-
son with the case where Mf, = and represents a useful factor from this point
of view. Therefore, to improve the flutter characteristics of a rotor it is
possible to use friction dampers in the feathering hinges. Of course, the use
of such dampers is possible only when the helicopter has a sufficiently powerful
and reliable booster control.
413
Section 4« Rotor . Flutter , yrith , Co nsideration of Coupling of
Blade Vibrations through the Auto matic Pitch Control
1. Forms of Rotor Flutter Observed in H elicop ter Experiments
As mentioned above, the occurrence of flutter in a helicopter sets up vibra-
tions of all rotor blades. These oscillations begin simultaneously despite the
fact that the parameters of individual blades making up the rotor generally
differ somewhat. Consequently, the simultaneous occurrence of flutter cannot be
explained by the coincidence of the
critical rpm of individual blades.
Furthermore, it has been noted in almost
all experiments on helicopters that the
vibrations of all blades are strictly
synchronized so that each advancing /383
blade duplicates the motion of the re-
treating blade with some lag in time.
The vibration amplitudes of different
blades increase simultaneously so that
their magnitude on the different blades
is approximately identical. Flutter of
one individual blade of the rotor of a
given helicopter is practically never
observed.
Plane of longitudinal
control
Swashplate of
automatic
pi tch control
Fig. 4. 21 Diagram of Rotor Hub.
This type of vibrations in flutter
is ascribable primarily to the coupling
of individual rotor blades through the
automatic pitch control (Fig. 4. 21),
The vibration mode of the rotor in
which each advancing blade duplicates
the motion of the retreating blade with some lag in time is usually called
cyclic vibration mode. Such modes are very often encountered in studies of
helicopter rotor vibrations. Therefore, they should be examined in greater de-
tail.
2. Analytical Expression for Cyclic Mode s__of Rotor^ Vibration
For cyclic modes of flutter, distinguished by the fact that each advancing
blade duplicates the motion of the retreating one, we can construct an analyti-
cal expression determining the law of variation of the blade motion parameters
in time.
If we fix the point of reference in time such that for t = we have Pni=o =
= 0, then this expression can be written in the following manner:
pAr= Po«" sin (p^ - A^A'p J,
(4.1)
where
Pig = flapping angle of the n-th blade;
414
go = angle determining the magnitude of blade deflection at the initial
reference time, for t = 0;
q = exponent determining the time rate of change of vibration amplitude;
p == frequency of oscillations in flutter;
Ai^B ~ phase shift of vibrations for two successive blades.
Equation (4«l) is used for determining the motion of blades with ramibered
N = 0, 1, 2, ..., Zt, - 1 (zb being the number of blades of the rotor).
For a blade with N = z^, the law of change of variables should coincide
with the law of motion of the blade having N = 0. Proceeding from this assump-
tion, the phase shift Ai|rm should be a multiple of the azimuth angle between /384
the blades, i.e.,
b
At critical flutter rjm, for q = 0, the vibrations of all blades take place
at constant and identical amplitude but with different vibration phases. The
analytical expression for the law of change of variables at critical flutter
rpm can be obtained by substituting eq.(4.2) into eq.(4«l) and setting q = 0:
^N=%^m{^pt~Nm^y (4.3)
It follows from eq.(4.3) that the vibration phase distribution for blades
in cyclic modes may differ depending on the quantity m. The quantity m is
called the order of the vibration mode and may vary from m = to m = z^, - 1.
At m - z^, the vibration mode of the rotor, as follows from eq.(4.3), will coin-
cide with the mode having the order m = 0. In like manner, for m > z^ all modes
will be repeated.- Thus, for any rotor there can be Zj, different vibration modes
corresponding to different orders m varying from m=Otom=Zt -1.
Equations (4*1) and (4.3), derived above for determining the modes of rotor
vibration, were constructed only for the variable Bn • However, all other para-
meters characterizing blade motion vary in the same manner. Nevertheless, a
certain vibration phase usually exists between them and the variable P^ • There-
fore, in many cases it will be convenient to represent the law of change of
variables in a complex form. With respect to the variable Pn > this can be
written as
p^=?„e"-'^^*'», (4.4)
where
It should be noted that, in forward flight of a helicopter, the blade exe-
cutes also forced vibrations of cyclic modes since, in flight, each advancing
blade duplicates the motion of the retreating blade. However, unlike vibrations
in flutter, the forced blade vibrations in flight are strictly synchronized
relative to the rotor rpm, so that each harmonic of vibrations of an order m
415
II
will correspond to the vibration mode having the same order:
?j,= ?^sinm(iot-Nf-y (4.5)
Here, m corresponds to the order of the harmonic of forced vibrations.
3. Cyclic Vibration Modes in Specific Gases a nd Con tr ol L oads
The division of vibrations into cyclic modes is convenient in that only
certain rotor control loops are loaded in the presence of each such mode.
Therefore, the critical flutter rpm is determined by the rigidity of that con-
trol loop which is loaded in the presence of the particular vibration mode under
consideration. Of practical interest are only those rotor vibration modes /385
that correspond to the smallest control rigidity and hence to the lowest critical
flutter rpm.
Let us study the manner of generation of cyclic vibration modes during
flutter, in a specific case - for example - for a four-blade rotor.
With a vibration mode of zero
identical phases and load only the
Position of coning
axis daring
flutter
Position of coning
axis before onset
of flutter
P
Fig. 4. 22 Position of Coning
Axis in Antiphase Flutter.
usually called antiphase flutter.
order (m = O), all four blades vibrate with
collective pitch control. This form of
flutter is called in-phase flutter. The con-
trol rigidity referred to the axial hinge,
and hence the critical in-phase flutter rpm,
depend only on the rigidity of the collective
pitch control loop.
The vibration mode of the first order
(m = 1), just as that of the third (m = 3),
is of greatest interest since on helicopters
it corresponds usually to the smallest con-
trol rigidity and hence to the lowest values
of critical flutter rpm. Vibrations of these
modes are characterized by the fact that only
the moment loading the lateral and longi-
tudinal control loops is applied to the
swashplate of the automatic pitch control.
The opposite blades in modes of the
first and third order oscillate in opposite
phases. Therefore, this mode of flutter is
The coning angle of the rotor in antiphase modes of flutter does not change.
Therefore, the motion of the blades in these modes is conveniently character-
ized by the motion of the coning axis (Fig. 4. 22). In vibrations of the first-
order mode, the cone of the rotor is deflected relative to the original axis
through an angle 3 and rotates about it with an angular velocity p^^ = Pfiu ~ '^
opposite to the rotor rotation.
416
Both the direction and magnitude of this angular velocity vary in the
third-order mode: pa = Pfiu + ou.
The vibration frequency of the variable forces in nonrotating parts of the
control system, just as the vibration frequency of the fuselage during flutter,
coincides in magnitude vri-th the angular velocity of rotation of the coning axis,
which constitutes the basic difference between these modes.
If the dynamic rigidity of the nonrotating parts of the control did not
depend on the frequency of forces applied to it, then the values of the critical
flutter rpm corresponding to modes of the first and third order would be identi-
cal. However, in all experimental investigations of flutter, only vibrations of
one of these modes, most often of the third-order (m = 3)> sre usually en-
countered. In several cases, in particular when the control system includes
inertia dampers, the first-order vibration (m = l) is observed in flutter. This
is explained by the fact that the dynamic rigidity of the nonrotating part of
the control, operated by the inertia inherent to its components, depends on the
vibration frequency. Consequently, the hinge control rigidities corresponding
to modes of the first and third order on a helicopter differ somewhat in magni-
tude. Accordingly, the critical flutter rpm also differs. These considera- /386
tions will be supplemented in Section 8.6.
m=0
m = /
m=2
N'o H\\2
N-t
N-2
N-3
n
/
\.
/
\
f
\
/
\j
1
\
A
^
n
\ 2
/I
\
M
L
\
m=
-3
r
V^
1
\
\^
I
\
M
/
%
/?
^
^
^'J
/
\
'\
J
\
J
f
Fig. 4. 23 Vibration Phase Distribution in Different Modes
of Flutter, for a Four-Blade Rotor at -^ = 1.75.
Droring second-order vibration modes (m = 2), the opposite blades in each
pair have an identical phase, and the phases of these pairs differ by half a
period. The forces applied to the control during vibrations of this mode are
locked on the swashplate of the automatic pitch control whose rigidity mainly
determines the hinge control rigidity for this case. Since this rigidity is
usually sufficiently high, the possibility of flutter with this mode, which is
usually called the plate mode of flutter, is improbable within the operating rpm
of the rotor.
417
The curves (Fig. 4.23) plotted on the basis of eq.(4.3) permit judging the
character of the phase distribution by blades in all these modes for a four-
blade rotor.
4. Differential Equations of Rotor Flutter -with Consideration
of Coupling of Blade Vibrations through the Automatic
Pitch Control "
Each rotor blade, during vibration, generates a moment acting on the blade
pitch control system. The magnitude of this moment, taken relative to the
feathering hinge axis of the hub, can be written in conformity with eqs.(2.19) as
where N = 0, 1, 2, ..., z^ - 1 is the numeral of the blade.
Here we have used the same notations as those given in Section 2 in deriv-
ing the differential equations of flapping flutter of an isolated blade with
axial flow past the rotor. Now, the number of equations has increased z,, times,
i.e., as many times as there are blades in the rotor.
If oscillations of individual rotor blades are in no way related and if /387
each blade is attached to the hub as an isolated entity, then, after substitu-
tion of
^C = c«;,.('-Pyv+v-P;v). (^^y)
into eq.(4.6), we obtain equations coinciding with eqs.(2.19).
However, helicopters usually do not have such rotor designs.
Generally, as a consequence of interference, the elastic angle of rotation
of each blade in the feathering hinge
yN=<fN-\-^^N (4.8)
follows the deformations of individual rotor control loops, which in turn are
determined by the totality of forces arising from all rotor blades.
For the conventional rotor control system, this relation can be represented
in the form
Y/v=Y..^ +Y.sin^l^» + Y,cosi.L^>+Y:,^>, (4.9)
where, as before, N = 0, 1, 2, . . ., Zi, - 1.
Here,
Yep - angle of rotation of the blade according to deformations of the
collective pitch control;
418
Yx and Yz ~ amplitude values of the angles of blade twist as a consequence
of defonnations of the lateral and longitudinal controls re-
spectively;
yI^p "= angle of rotation of the n-th rotor blade as a consequence of
deformation of the swashplate of the automatic pitch control
under the effect of forces completely balanced on the plate;
it is assTjmed that, if all external forces are balanced on the
swashplate, its deformation obeys the condition
2:vr;=o; (4.10)
N
•a.p
where t^^^ is the azimuth of the N-th blade reckoned from the plane of the
longitudinal control with respect to the swashplate spider (see Fig. 4. 21); this
azimuth is related with the blade azimuth by the expressioh
where
Qgt = angle of stagger of the rotor hub spider;
Ate on ~ control angle of advance;
I = blade angle of lag during rotation about the drag hinge; in
Fig. 4- 21 the blades are shown in a position where 5=0;
,(.V) ,(0) I »r2Jt
If the rotor has three or less blades, then the quantity Ya!*? should be
set equal to zero, since in this case there is no combination of forces which
could be balanced completely on the swashplate.
For a foixr-blade rotor, all values of yI^p are equal in modulus, i.e., when
N = 0, 1, 2, 3
lv^'l = --^- (4.12)
This equality is not observed for a greater number of blades. /388
If we introduce the concepts of rigidity of various control loops referred
to the axial hinge of the blade, then the hinge moment acting on the blade due
to the control can be expressed in terms of these rigidities and deformations
of the corresponding control runs:
^^^ol = <=.p yep + ^xY. sin ^%' + c,Y. cos i>l^' + c^^vl,^', ( 4. 13)
where c^.p, c^, c^ , and Ca.p are the rigidities of the collective pitch control,
lateral and longitudinal controls, and swashplate respectively, referred to the
feathering hinge of the blade.
a9
The form of notation of eq.(4.13) assumes that the'rigidity c^.p remains
constant regardless of the type of combinations of forces locked on the swash-
plate.
The values of the deformations of different control loops referred to the
feathering hinge can be expressed in terms of the angles of rotation of indi-
vidual blades Yn i^ "^^ represent eqs.(4»9) as a system of equations relative to
the unknovms Yc,p» Yx > Yz > ^^^ Ya.p* The solution of the system (4«9) yields
the following expressions for deformations of individual control loops:
c.j,-—^yN,
N
When
^ N
N
(4.14)
,(N)
For a four-blade rotor, Va.p can be detemiined by the formula
'a.p
■ COS nN
2y.
cos nN.
(4.15)
For a number of blades z^ > 4> "the quantity Ya'!p is determined by the
expression
^'"7==}- S cos^AT^YyvCos.
2nm
N.
(4.16)
m=2
N
Substituting eqs.(4.l4) and (4.16) into eq.(4.13), we obtain the expres-
sions for the hinge moment from the control for a blade with the numeral N:
fCA')— 1
+ — c,cos^w2y^cos.)-^^) +
N
(4.17)
+-
N
Substituting eq.(4.17) into eq.(4.6) and examining this equation to- /389
gether with the first equation of the system (2.19), we obtain a system of dif-
ferential equations of coupled blade vibrations at axial flow past the rotor:
420
where
-i-^ Pi Sin ^(^n 2 (cpyv + -^n) sin ^.<^^) +
* N
+ 7" /'z cos 1-^^) 2 (^^ + ^^^^ ^os -I'L'^; +
* . N
+ 7-^^/ 2 cos-2^ArV(cp^+.p^)cos^^Ar=0.
/n=2
(4.18)
(4.19)
The system of equations (4.18) is a system of ordinaj?y differential equa-
tions relative to the unknown functions P^ and cpfj , with periodic time-variant
■coefficients together with the variable
where
^iN) = ,,t-Ni,%,
(4.20)
^%-
5. Transformation, of Eqs.( 4.18) in Particular Cases where
Cyclic Modes are the Solution of the Differential
Equations of Rotor Flutter
Let us check whether cyclic vibration modes are the solution to the dif-
ferential equations (4.18) of rotor flutter written with consideration of
coupling between blade vibrations through the swashplate.
In the general case, the relation between variables in cyclic vibration
modes of a rotor can be represented in the form
(4.21)
421
where /390
3o and cpo = angles of rotation of the blade with the numeral N =
relative to the flapping and feathering hinges, which are
P unknown functions of time;
Mb ~ — 2 — ^ phase angle characterizing the vibration mode of the
^ order m.
Substituting eqs.(4«2l) into the differential equations (4. IS) and succes-
sively varying the values of m from to Zi, - 1, we find that cyclic vibration
modes are the solution to eqs.(4.18) only for values of m = (in-phase flutter)
and z - 2 s m s 2 (plate mode of flutter). At these values, the differential
equations (4. IS) are transformed into equations exactly coinciding with the
equations of flutter of an isolated blade [eq.(2.19)]. Only the value of the
frequency of natural vibrations of a blade in torsion entering the second equa-
tion of system (2.19) becomes equal to
during in-phase flutter (m = O) and
P,„-Pc.p (4.22)
Pf^ -Pa.p (4.23)
during plate flutter (z^ - 2Sms 2), when all forces due to the blades close
to the swashplate and the lateral and longitudinal controls and collective pitch
control are not loaded.
At the same time, cyclic vibration modes at m = 1 and m = z^ - 1 are the
solution to the differential equations (4. IS) only in one particular case, when
Cx = Cj . In this particular case, the differential equations (4. IS) are trans-
formed into equations coinciding with eq.(2.19) for an isolated blade. Only the
value of pt« in this case should be equal to
P^^=Px=pz. (4.24)
Thus, flutter of a rotor as a whole can be studied on the model of an iso-
lated blade having a rigidity of attachment equal to the rigidity of the col-
lective pitch control Co.p, with the cyclic pitch control c^ = c^ and swash-
plate Ca. p taken separately.
6. Rotor Flutter in the Pre^ence_ of_D ifferent Rigidity
of Longitudinal and Lateral Cont rols
To solve the differential equations (4. IS) in the case of c^ 7^ c^ , we can
use the following method. Let us introduce the new variables:
N
N
(4.25)
422
N
Successively multiplying all terms of eqs.(4.18) by sin i|ri^p and by /391
cos ij'i'llp and summing them with respect to N, we obtain a system of ordinary dif-
ferential equations relative to the new variables of the following form:
-f (02 (6,2 — r,2) ?y — 2 C,20)T1^ — rf,2U)'Tl,p = 0,
Tip 4-flrnU)T]p + 2(Bep +rf,iU)2$p -|-Ci2Ti,.+ di2<or\f +
-f 0)2 (6,2 — C,2) T]^ + 2C,2(0;*^ + C?,2«)% = 0,
r„ + ar22<4+[(622-l)'»' + /'^]?f-2o/T)^-flr22«>^Tl,+ ^ (4*26)
+ ^21^3 + ^2i">-.p + /'^'^'P — 2c2i(OTip — d2yy\? =0,
% + rf22«% + [(^22 - 1 ) t"^ H- PI] nf + 2 0)if + d2^<0% +
+ r2iT,p+Cl'2,(UTlp+jD2xTl3 -J- 2C2,U)';p^^C?2i'»^^?=0.
The system of equations (4.26) can be solved by the conventional method
for solving a system of differential equation's with constant coefficients.
A similar method of reducing the problem to a system of equations with con-
stant coefficients was used by Coleman and B.Ya.Zherebtsov in investigating the
ground resonance of helicopters.
We can show that the variables (4.25) can be expressed by the variables
proposed by A.P.Rroskuryakov for investigating helicopter stability.
In his works, A.P.Proskuryakov expressed the angle of rotation of the blade
relative to the flapping hinge in the form
P;v= Oo (0 + «i (0 cos .]j;v+ ^ (0 sin tl-w.
(4.27)
On alternately multiplying eq.(4.27) by cos ilr^ and sin 1];^ a^nd summing with
respect to N, it will be found that
a,(^)=— V,pArCOs<l<;v.
N
(4.28)
i.e., the variables ai(t) and bi(t) virtually coincide with the variables Tig
and I3 .
423
The use of the above method for solving equations of helicopter flutter at
Cj f Cj and iJ. 7^ was proposed by L.N.Grodko. It was also used by V.D.Il'ichev
for obtaining practical results.
Section 5. Flapping Flutter of a Rotor in Forward Flight
1. Preliminary Statements
Experiments carried out on various helicopters showed that, in forward
flight, flutter might set in earlier than under conditions of axial flow past the
rotor, for example, in ground-testing. Therefore, a determination of the /392
critical flutter rpm in flight is of appreciable practical interest. The basic
problem requiring solution in this case is the degree to which the critical
flutter rpm is lower in flight than on the ground.
A variety of other important practical problems arises in this connection.
For example, what parametric margin prior to flutter should be secured under
ground-testing conditions so as to preclude the possibility of the occurrence of
flutter in flight.
All these problems can be solved if there is an opportunity to calculate
flutter in forward flight, which permits determining, in particular, the de-
pendence of critical rpm on the flying speed.
Furthermore, difficulties arise in calculating the flutter in flight.
These refer primarily to substantial complication of the differential equations
describing blade Aribration. Therefore, in examining the problem, one should
begin with these.
2. Differential Equations of Blade Oscillations
in Forward Flight
The differential equations of torsional and flapping vibrations of a blade
in forward flight are derived in the same manner as for the regime with axial
flow past the rotor. Only the values of the relative velocities of the stream
flowing past the profile should be calculated with consideration of the addition
term due to forward velocity. These velocities can be written in the form
i/^ = cD;?(7+(isin<)<), j
, V cos 01
^here V' = —^
Substituting eqs.(5.l) into eqs.(l.6) and then eq.(l.6) into eqs.(2.3)
and (2.11), we obtain the differential equations of blade vi.br5.tion in forward
flight:
424
P + Ki + l**!, sin<l.)u,p+^l+|x&|Jcos<ji4--L(J,2ft^si^2<l>^u)2p-
1 1 ^ - _
^ P-^^ii (1 - cos 2^) (o2(p = o>2yo J ft?X (r+ [J, sin <|>) dr,
-*
- cos 24,)j a)2 + ;,^2 J <p + C2,^'+(rf2i -(ifti^ sin ^)(op +
4- ([^2, - (iftlj cos <}- - -i- (1,2 6^1 sin 2i}.l U)2 + y.;7^2 j p ^ ^2 9 _^
+ 7^ J ^mA?^,„^ rfr - (o2 ^ J 6^ X (F4- [X Si n ^) cfr.
Here, the coefficients Cis, cg^ , d^ , dgi , dgg , bgi , bgs are the same
as in eqs.(2.14); furthermore, we introduce the following additional coeffi-
cients:
(5.2)
/393
"22
T^^/irl*"^^'--
(5.3)
Assuming that the particular solution B'^ and cp"*^ of eqs.(5.2) is found, we
set, as before (see Sect. 2. 4)
where B^ and cpa are the angles of deflection of the blade from a position
425
corresponding to its steady motion, determinable by the particular solution.
Substituting these expressions into eqs.(5.2), we obtain the follovdng dif-
ferential equations of disturbed motion of the blade in forward flight:
? + ('^n + P'*!! sin <!)) (Op + ( 1 + |j.*}j cos ([) ^- — (i.26| I sin 2(1)) o)2p -J-
+ ^12? + K2 + t^of !2 sin <1)) (o<p +
+ [*i2 — i-[^6i,sinil>--l-p.26|{(l-cos2.1,)lu,2tp=0,
9 + (0^22 + H'C'^z sin '^) 0)9 + jr^jj + -i- [xftij sin <}- +
-r-f l^'ftlHl-cos2<}.)]<o2+;72 j9 + C2iP+(rf2i-pL6>,sin<i;)(Bp+
f { 62,-ix6i2COS(l>-^[*2*^>sin2.;-l«)2+x/7^2 h=0.
(5.4)
Here, the subscripts of the variables p^ and cp^ , which indicate that they
refer to disturbed motion, are omitted for simplicity.
3. Solution of Differential Equatio ns /394
Equations (5*4) represent a system of differential equations with periodic
coefficients. The solution of such a system can be -written in the form
(5.5)
where the functions Tg and T„ determine the content of the harmonic components
of blade vibration in flutter.
These functions can be written as
^P = 2 (Pn cos rt^ + P„ sin «<)-);
n
^f = 2 (?« cos «<1< + 9„ sin raip) ,
(5.6)
where n = 1, 2, 3, ... are constant coefficients determining the order of the
corresponding harmonics.
The critical flutter rpm in this case can be determined if eqs.(5.5), with
consideration of eqs.(5.6), are substituted into the differential equations (5.4)
and if the coefficients of like harmonic components are equated. This operation
results in the formation of_a system of algebraic equations relative to the un-
known coefficients Po , 9o > 3n » Pn » 9n » and cp„ . To solve this system, it is
necessary to determine the roots of the characteristic equation whose order
426
depends on the number of harmonic components n retained in the solution.
The solution of eqs.(5.4), with consideration of the harmonics, greatly
complicates the calculation and at the same time - at the values of (j, < 0.4
actually used - introduces no essential refinements into the calculation results.
Therefore, in practical calculations we usually employ either the approximate
method without consideration of the harmonic components or else the method of
calculation with numerical integration of the equations of blade motion with
respect to time. One of the versions of this method will be given in Section 7
of this Chapter.
4. Determination of Critical Flutter Rpm without Consideration
of Harmonic Components of Blade Motion
If the effect of harmonic components on the critical rpm is disregarded,
the calculation of flutter in forward flight is no more complex than under con-
ditions of axial flow past the rotor. An approximate solution, neglecting the
effect of harmonic components can be obtained, if the periodic coefficients in
the differential equations (5.4) are omitted"'. In this case, the forward flying
speed is taken into account by introducing, into eqs.(l.6), the constant part of
the functions depending on Ux . For this it suffices to set
U2^^^2/^2fr2-^l-^-^y j (5.7)
Then, the system of differential equations of disturbed motion can be /395
written in the following manner:
'f + '^22""f + [(jf +">') + *22 (l - Y t^'*2*2 ) »' 1 f +
(5.8)
where
^^2 =
^12
II
*22
For a blade of rectangular planform, the coefficient bfg can be considered
as approximately equal to -2. The coefficient bgg is small in magnitude and has
no substantial effect on the results.
Equations (5.8) differ from eqs.(2.19) for a regime with axial flow past
the rotor only by terms of the type of f 1 — • \i^hfs)' This permits determining
* This method was proposed by V.D.Il'ichev.
427
the critical flutter rpm in forward flight by eq.(2.27); -however, in the expres-
sions of certain coefficients of eq.(2.24) entering this formula there appears
an additional term of the type f 1 — [i^'b^s)'
— *12C2l[l — — (^^^^2) — ^=^12 0^21— ^12^21],
(5.9)
Thus, disregarding all harmonic components of blade motion, the problem of
determining the critical flutter rpm in' forward flight .can be reduced to solving
the system of differential equations (5.8) with constant coefficients.
5. Effect of Flying Speed on Critical Flutter Rpm
/ 1 2 ->'-
The effect of flying speed, definable by the term II — • p, bi-
m
eqs.(5.8) proves to be quite weak. Figure 4.24 shows the dependence of the
critical rpm on the flying speed, determined by the value of p,, for three dif-
ferent values of blade balancing.
If follows from the graph (see Fig. 4. 24) that the critical 'flutter rpm /396
drops by about 5 - 10^ with an increase in
flying speed to values of |j, = 0,25 - 0.3.
In experiments carried out on heli-
copters, the effect of speed is somewhat
stronger. This can be explained by the ef-
fect of the following factors:
It is shown above in Section 3 that,
for small blade oscillations during flutter,
the axial hinge with friction can be re-
garded as a linear damper whose efficiency
is smaller, the higher the angular velocity
of relative displacements in this hinge
during forced vibrations of the blade.
Therefore, the critical flutter rpm in
flight decreases with increasing relative
'*/lu
/
1
f
'
^-F./r^*%
nt
■-^
x,„'25'/.
^
-^
-
""
^m-^n
—
^
;;;^
— 1
0.Z
««
Cf
as
Fig. 4. 24 Critical Flutter Rpm
as a Function of Flying Speed.
428
displacements in the axial hinge and hence with flying speed, since relative
displacements usually increase mth speed. Hence it follows that all factors on
which the helicopter balancing depends may affect the flutter, since balancing
determines the vibration amplitude in the axial hinge with respect to the first
harmonic of rotor rpm.
Displacements of the blade in the axial hinge, with harmonics higher than
the first, may also have a strong effect. These harmonic components usually
have smaller amplitudes of displacement but relatively high angular velocity,
leading to an appreciable reduction of the effectiveness of the damping action
of dry friction in the axial hinge of the blade.
Thus, in many cases the severe drop in critical flutter rpm in forward
flight is explained by a decrease in the damping action of friction in the axial
hinge .
A no less important factor capable of substantially influencing critical
revolutions of flutter is the variation in the aerodynamic characteristics of
the blade profile in connection with fluctuations of the value of the Mach number
under forward flight conditions. As mentioned above, a change in M in the range
from 0.5 to 0.9 causes a marked change in the aerodynamic characteristics and,
what is especially important for flutter, a distinct shift in the position of
the profile focus.
Only the method employing numerical integration of the differential equa-
tions of blade motion with respect to time (see Sect. 7) permits taking into
account these factors with sufficient accuracy.
Section 6. Calculation of Flutter with Consideration of
Bending and Torsion of the Blade
1. Bend ing and_Torsion of Blade during Flutter
It was pointed out above that, in the overwhelming majority of cases, vi-
brations of the blade as a solid body predominate in the mode of blade vibration
in the flapping plane during flutter. The blade executes these oscillations,
rotating about the flapping hinge. Torsional vibrations of the blade occur
mainly as a consequence of its rotation about the feathering hinge. In this /397
hinge, the blade rotates owing to the kinematic action of the swashplate of the
automatic pitch control and flapping compensator as well as deformations of the
control cables. Flexural and torsional deformations of the blade itself general-
ly have no significant effect on the critical flutter rpm. Nevertheless, the
flexural and torsional deformations of the blade during flutter of this type are
usually quite pronounced. They lead to smaller displacements of the blade ele-
ments in comparison with displacements during vibration of the blade as a solid
body, but these displacements are of the same order. Therefore, it is impossible
to neglect deformations of the blade itself or to show no interest in them.
In individual cases, the flexural deformations of the blade increase and
begin to have a noticeable effect on the critical flutter rpm. It is especially
important to take into account blade bending in determining the effect of con-
429
centrated balancers installed on the blade to eliminate -flutter.
Also known are individual cases where the blade during flutter executes
flexural vibrations in which the share of the flapping mode is quite small. It
should be emphasized that such cases are very rare. However, for jet helicopters
with blade-tip engines, such flutter - usually called "bending flutter" - con-
stitutes a serious danger. Subsection 8 of this Section will be devoted to an
examination of this type of flutter.
As stated above, the effect of torsional vibrations of the blade during
flutter can be disregarded at a value of the coefficient a < 0.4 - 0.5 (see
Sect. 1.4). In the remaining cases, in particular when the pitch control system
has relatively great rigidity, blade torsion cannot be disregarded. This may
result in a very large error..
However, most of the presently constructed helicopters have a coefficient
a < 0.4* Therefore, in Subsection 6, we will specifically study flutter with
consideration of bending but without consideration of torsional deformations of
the blade. Such an approach leads to a considerable simplification of the dif-
ferential equations.
2. Determination of the Torque from^ ending Forc es on. the Blade_
In calculating torsional strains of a blade it is important what method is
used for determining the torque due to bending forces on the blade. If the
blade is bent in the flapping plane, then the force Q applied to the blade in
the plane of rotation creates torque on the arm Ay relative to the section, at a
radius r closer to its root (Fig. 4.25). Likewise, when the blade is bent in the
plane of rotation a similar
torque on the arm Ax is created
by the force T acting in the
flapping plane.
In calculating the twisting
moments due to bending forces
on the blade, it is important to
recall the fact that the com-
ponents of the centrifugal
forces relieving the blade in
bending also participate in the
generation of twisting moments.
If we calculate only the torque
due to external bending forces
on the blade, the value will be
much larger than the actual
torque, just as the moment due
only to the external forces bend-
ing the blade will be many times greater than the bending moment in the blade
section.
Let us examine a blade element of length dr, bent in two mutually perpendic-
Fig.4.25 Diagram of the Occurrence of
Twisting Moments due to Bending Forces
on the Blade.
430
ular planes (Fig. 4- 26), Equating to zero the sum of the moments of all external
forces relative to the tangent to the blade axis in a section at the radius r /398
and discarding all terms of higher orders of smallness relative to dr, we obtain
-Mt„+Mf^^dM,^-M,^!'dr^M,y"dr=Q (6-1)
or
dr
-.M^'-M^y". (6.2)
If, for simplicity, we assume that the planes of maximum and minimum blade
rigidity coincide with the planes of rotation and flapping, then, having set
and
we obtain
V'=^ (6.3)
My
S'--^. (6.4)
'-^-"Mir.-ii;)- f^-''
where I^ and ly are the elastic moments of inertia of the blade section during
bending in the plane of rotation and flapping plane.
Equation (6.5) was first proposed for calculations of a blade by V. N.Novak.
It follows primarily from an examination of this formula that the torque
BMtor
— ^ per unit length due to the bending forces on the blade is always equal
to zero if
(6.6)
i.e., if the rigidity of the blade in the plane of rotation and in the flapping
plane is identical.
Furthermore, by virtue of the smallness of the bending moments Mx and My
(as a consequence of load relieving by centrifugal forces, these moments are by
a factor of 8 - 12 less than the moments due to the external forces acting on
the blade), the torque — ^°'' per unit length will be quite small in all cases
even if I^ 7^ ly. This conclusion is highly important and results in a general
approach to calculating torques and torsional deformations of a blade, as
follows :
In each section of the blade, we must determine the torque relative to /399
the flexural axis of the blade in the examined section due to forces acting only
in this section. Then, these local twisting moments should be summed with
431
Fig. 4. 26 Diagram of Loading the
Blade Element with Stresses in Two
Mutually Perpendicular Planes.
respect to the blade length. Hence it
follows in particular that the arms of
the forces causing the twisting moments
of the blade must remain constant re-
gardless of whether or not the blade is
bent.
With regard to flutter calculations,
it follows from this conclusion that the
torque per unit length of the blade from
centrifugal forces should be calculated
by the formula
dMt..
SL=m2
dr
= ia'^mray\
(6.7)
rather than by the frequently used
formulas of the type
or
-<sy^ y' ( mradr\
(6.8)
(6.9)
which holds true only for a blade with an infinitely great rigidity in the plane
of rotation.
3. Differential Equations of Binary Blade Vibration
Binary blade vibrations in vacuum are examined in Section 5, Chaoter I of
Vol.11. In studying binary vibrations in air, we must additionally take into
account aerodynamic forces. Using the differential equations of flexural
[eq.(l.9)] and torsional vibrations [eq.(5.6)] of a blade (see Chapt.I of /400
Vol.Il) and supplementing these with inertia terms of the couple and with aero-
dynamic forces expressed by eqs.(l.6) of this Chapter, we obtain a set of dif-
ferential equations of torsional blade vibrations in air:
my + [£//]" -lA^i/'l'-macp-
- i- clQb \U\ + (^* - ^o) U!f - Uy]^ = 0;
/mT-[Gr«']' + <oV„cp -l-^e6^f/? +
+ \clQby [f^'?+(-f- *-Jro)f^?-^i]-<«''«»'-i/' -'«=i/=0.
(6.10)
432
These equations are -written in a form pertaining only to disturbed motion
of the blade. The particular solution describing undisturbed steady motion of
a blade will not be discussed here.
In eqs.(6.10), we use the following designations:
y = displacement of the blade element in the flapping plane during
disturbed motion of the blade;
cp = angle of rotation of the blade element in the same motion;
m = mass of the blade element per unit length;
I„ = moment of inertia of the blade element per unit length relative
to the feathering hinge axis;
GT = torsional rigidity of the blade;
N = centrifugal force in the blade section:
A^ = (o2 f mrdr;
a = distance from the center of gravity of the section to the feather-
ing hinge axis, with the direction from this axis to the trailing
edge of the blade considered positive;
a^ = distance from the profile aerodynamic center to the feathering
hinge axis.
Differentiation with respect to the blade radius is denoted by a prime and
with respect to time by a dot.
To solve this set of equations it is convenient to change from the variable
cp, which determines the total angle of rotation of the blade element in dis-
turbed motion, to the variable ■& representing only the elastic angle of rotation
of the blade and correlated with cp by the relation
cp = & — xi/o.
where
Jq = angle of rotation of the blade in the flapping hinge;
H = flapping compensator.
Let us substitute the expression for the angle cp into the differential
equations of binary blade vibration [eq.(6.10)]. This makes it possible to /401
rewrite them in a form more convenient for further transformation:
my^[EIy'T - [Ny-\ + mo^y, +
+ -1 c^qb^ [fy^^o +(t '' " -^o) ^^"] "
(6.11)
433
— may — ijy'o — w^I„xy'o ^ Qb^Uxy'o —
In the presence of a horizontal flying speed of the helicopter, the rela-
tive velocity of the flow past the profile will be a periodic ftmction of time
and radius. This velocity can be set approximately equal to the velocity Uxt
6^=u)r+Vsin<Bf.
(6.12)
Therefore, eqs.(6.1l) represent a system of partial differential equations with
coefficients periodically varying in time.
When the flying speed of the helicopter V equals zero, the periodic coeffi-
cients of the system (6.11) become constant, independent of time.
For the examined type of rotors system, eq.(6.1l) has the following
boundary conditions:
Afo = [£/t/"]o=x(A/, + A//r).
(6.13)
where
M© = bending moment in the blade root;
Mt = twisting moment in the blade root;
Mfr = moment of friction in the axial hinge of the hub;
Ccon ~ rigidity of the control system;
«?o = angle of rotation of the blade root due to deformations of the
control system.
4. Solution of Differential Equations
The solution of the system of differential equations (6.11) can be obtained
by using B.G.Galer kin's method. We set
(6.U)
where
y^^^ and ■d^^^ = modes of the natural flexural and torsional vibrations of
the blade in vacuum;
6j and Yk ~ coefficients of flexural and torsional deformations of the
434
blade with respect to the j-th flexural and k-th torsional harmonic
of natural vibration.
The coefficients 6j and Yw aj^e certain functions of time. Since eqs.(6.1l)
are differential equations vrith periodic coefficients, the coefficients 6^ /402
and Yk should be functions of time of the type
S^=8^e"(l+7') (6.15)
where the function T determines the content of harmonic oscillations during
flutter.
If, as before in Section 5*4, we seek the solution with an accuracy limited
only by the fundamental frequency and disregard the effect of harmonic components,
then we can omit the periodic coefficients in eq.(6.1l).
Applying B.G.Galerkin's method to this simplified system of equations, we
obtain a system of ordinary differential equations relative to the variables 5^
and Yk • In matrix form, this system can be written as before (Sect. 2. 5) as the
equation
Here the variable X is the vector function with projections 6j and Yk > i.e..
^- " (6.17)
while A, B, C, and D are rectangular matrices of the order z, where z is the sum
of the number of flexural and torsional harmonics accounted for in the calcula-
tion.
Setting X = Xoe^ * in eq.(6.l6), we obtain a system of algebraic equations
of the form
\CX^ + D<.\^A + o,^B\X,=0. (6.18)
Let us then equate the determinant of this system to zero. The resultant
algebraic equation relative to the unknown parameter X is the characteristic
equation of the system (6.16). The roots of this equation completely charac-
terize the blade motion described by the system (6.11).
To determine the boundaries of flutter, we should set A. = ip in the charac-
teristic equation and find the corresponding values of cu and p. These values
will determine the parameters of the limits of the flutter zone.
435
w
An analysis of the results obtained from calculations shows that, in the
general case, each combination of torsional and flexural harmonics of blade vi-
brations may correspond to a zone of instability vjith oscillations having a mode
in which the content of the harmonics of this combination predominate. However,
with actually used blade parameters, a given flutter zone by no means corre-
sponds to each combination of harmonics. Thus, the nvmfcer of flutter zones is
always smaller than the number of combinations of flexural and torsional har-
monics and can never be greater than the number of these combinations.
For practical purposes, an important point is the direct dependence of the
critical flutter rpm on the frequency of the natural vibrations of the torsional
harmonic of the blade entering into the combination in question. Therefore,
combinations involving only the first harmonic of torsional blade vibration /403
give the lowest values of critical flutter rpm. All other combinations based on
higher torsional harmonics of the blade are of no practical interest since the
critical flutter rpm corresponding to these zones is always higher than the
operating range of interest here.
All forms of flutter, corresponding to combinations of different flexural
harmonics of the blade with the first harmonic of torsional vibrations of the
blade, will be called the principal modes of flutter. Below, we will be in-
terested only in the principal vibration modes since these modes of flutter have
the lowest critical rpm and therefore are the only onfes encountered in practice.
5. Calculation of Flutter with Consideration of
Three Degrees of Freedom
To illustrate the above method, let us examine in greater detail the compu-
tational formulas for the case where the vibration mode during flutter is repre-
sented as combinations of the zero r and the first y flexural and first tor-
sional harmonics.
The matrices entering eq.(6.l6) will be of the third order in this case,
and the vector function X will have only three projections:
Ar=
(6.19)
The coefficients of the matrices A, B, C, and D will be referred, as above,
to the values of the coefficients Ih.h* Li, and I^ standing for the higher
derivative of the variables:
/:, = J /JMr;
R
/i = f my'^dr.
' 1-
436
Let us write out the expressions for the coefficients of the matrices:
a) Inertia matrix C:
C =
where
^U ^12 <^13 \
Cji C22 C23 I '
^^31 ^32 ^^33 /
mar dr.
"-'+^J
mar dr,
c„ =
r22 = l,
R R -<
J
r23 —
1
« « -1
J
<^3i = y- I may dr,
R
b) Damping coefficient matrix D:
/flf,l ^12 rf,3
( "21 ^^22 ^^23
W3I '='32 "^33 .
where
11 2 «' /
A.A
Lo p J
""—■ J-'-tl'^^lT-f)'*-
(6.20)
(6.21)
/404
(6.22)
(6.23)
437
rf.3 =
R
Cinf —
*22
LO J
^1
23 2 " Z,
'A-^V^r
* /
--^Q-^PoJ*^^»rfr-x?„»„^,„
c/,,=-L^_^
2 '«' /,
(■/
Lo J
2a,
if dj. = , ^'' [see eq.(3.14)].
TrLxntpp r
c) Stiffness matrix A:
/O 0'
A=io a^^ I.
^0 Oct.
where
^33/
-r,^ =— ^
,2
V
«22 = /J
«33 = /'o
(6.23)
/405
(6.24)
(6.25)
Here, po, is the frequency of natural flexural vibrations of the first harmonic
of a nonrotating blade.
d) Centrifugal and aerodynamic stiffness matrix B:
438
i
where
*U *12 *13 \
^ — I b^i ^22 *23
\ *3l *32 *33
;■
*n=l+4
bn = Y cje^^ Po [j ft^' rfr + -I- H.2/?2 j 6r rfrl ,
*2i = - — f /raord rfr-1- X r fjb dr-\-
'1-0 J
''''^~^ [t '''^^''^° J *"' ^'^ '^'' + Y t^'^' f *°/ »'''■]
lo - J
k R
u
R K -I
r ftr^i/rfr + -1- ii.2/?2 r & J/ rfr ,
J
R -I
l-|x2/?2 Uyddr ,
J
R R -1
J
— C o
2 " /,
i!'32=-Y'^;77[|*^'^^'^"+
*33=*+^<^;e 7j-fo
(6.26)
(6.27)
A06
Here, k = ^= — rp^jmrdr where P = y'.
The characteristic equation for this case will have the following form:
XMo + X5a) A, + k< (B,(«2 + fij) + X«(« («)2C, + Cj) + X2 (a)«D, + oi^Dj + D3) +
(6.28)
439
Hence,
+ X(o(cO*£, + U)2£2+£3) + <»M'«*^l+«''^2+^3) = 0.
^ 1 == flr„5o + c„5i + flr,3/?o + cig/^i + 0^127^0 + <^i27'i ;
j9, = 6„5o + ^1 l-S"! -1- '^I A + *13^0 + ^13^1 + <^13'^2 +
^2 = CjjOs -|- C13/C3 "T ^12' g!
Ci = 6„5, + rf„52 + C„54 + 6,3/?! + rf,3/?2 + C,3/?4 +
+ *I27') + ^12^2 + ^127^4;
C2 = rfl,53 + CiiSg + rf,3/?3 + <r,3/?5 + rfizT'g + Cj27"s;
£>, = 6„5o + rf„54 + C„5, + *,3/?2 + rf,3/?4 + '^13''?6 +
+ *12^2 + ^127^4 + ^li^e'^
£>2 = *n53 + flfll'^s + ^^n^y + ^3^3 + ^13/?S + Cl3^7 +
+ *i27'3 + dn^s + Cl2T^•,
Ei = buS, + duS,-\-b,,/^, + di,R, + b,2r, + d,2Te-^
E2=^bnSs^dnS^ + b,^R,^d,^R^^b,2T,-{-d,2T^■,
Cz'^dnOfj,
F, = bnSe + b,sf^, + b,2Te;
f2=bnSj+bi3R.j-\-bi2Tj;
•'3 = ^11*^8'
(6.29)
where
■ C32U23'
Sq = C22C33 C23C32;
•J 1 = f'22"33 r ^33"22 <^23^32 '
•^2 = ''22^33 ~l" ^33*^22 4" d2^33 — £'23'^32 ~
63 =^ ^22*^33 ~r ^33'-22>
•J 4 ^^^ ^22"a3 ~r b 33d 22 — b^^d^i ''32"23i
•^s ^^ o-iid^ "T ^^33^22!
■ ^32'- 23 "
-rfo.,^;
23"32>
"J 6 — b<^^-
- b23"32-<
'7 -^ 022^33 "T <^33^22'
57
"J8^^^^22'^33'
^0 = ^21*^32 — '^22''3i;
/?, = ^21^/32 + ^32^21 — <^22^31 — ^aidii,
R2 — "2\^:
21'- 32
- ^32^21 "I" d2\dsi -^ ^22^^31 — ''3l'^22 "
■ Ofoorf.
22"31'
/?3 — Cl22^3\'<
/?4 = 601^32 + ^32'j'21 — 622^31 ~ *31^22'.
/^5= 022" 31'
^6 ^^ ^21*32 — ^22^31 !
/4O7
(6.30)
hhO
^7=— «22*3i;
^0= ^23*^31 ~ ^21^33;
' 1 = '^23"31 ~t" ^31" 23 ^21*^33 — <^33''21'
T2'=l^23C3l + *3lC23 + '^23^31 — ^21^33 "
" ^33*^21 ~
-d^.dsi.
'3= — ^33*^21;
T^ = ftjgC^a, + 631^23 — ^21^33 — 633^21 ;
T^= —a^d 2u
7'6=*23*31— ^21^33;
7^7=— «33*21
The roots of the characteristic equation (6.28) can be determined by means
of any standard program available for digital computers of any type. Such a
program can include the operation of computing the coefficients of the charac-
teristic equation directly from the coefficients of eq.(6.l6). In this case,
eqs.(6.29) need not be used.
The values of the angular velocity cu corresponding to the limits of flutter
can be obtained also directly if, in the characteristic equation (6.28), we set
X = ip and equate to zero the real and imaginary parts of the equation separately.
The equations thus obtained vri.ll have the following form:
^iC"', yt7) = ;'Mo-/j''(»25, + fl2)+/'M'«''£>i+"^'£>2 + -03)-
io(u,, p)=/7M, - p2(to2C, +C2) + (B'«f , +«,2£2 + £3 = 0.
(6.31)
(6.32)
If, from the equation L2(uu, p) = 0, we determine p = f(u}) and substitute
into the equation Li(ci), p) = 0, then the points of intersection of the obtained
curve Li(u)) = with the abscissa will correspond to the limits of flutter.
Calculation of Fluttgr__with Three Degrees of
Fr eedom Disregarding Blade Torsion
/AQ8
All the formulas presented above are appreciably simplified if we assume
that the rotor blade is absolutely rigid in torsion. It was noted above that
this assumption is valid for all rotors for which the torsional rigidity of the
blade is appreciably higher than the rigidity of the blade pitch control system.
In this case, during torsional vibrations the blade elements rotate mainly as a
consequence of deformations of the control system and, to a lesser degree, owing
to deformations of the blade itself.
Consideration of a variation in the angle of rotation of the blade with
respect to length leads to a minor change of certain coefficients of eq.(6.18)
[see eqs.(6.21), (6.23), and (6.2?)]. This is explained by the fact that the
magnitudes of the integrals entering the expressions of these coefficients are
determined mainly by the blade tip which is subject to large aerodynamic forces.
while the change in the angle of rotation s9 over the length of only the blade
tip is insignificant. Therefore, the assumption of constancy of the angles of
rotation of the blade cross sections over its length, in many cases, will not
lead to substantial errors. At the same time, this assumption appreciably
simplifies all computations, since i? = 1 and there is no need to decompose the
angle of rotation of each blade section into ?? and kj4 •
The differential equations of motion for this case can be written in the
following manner:
-\-{^b — x^U'<!}- Uy\ — moy\dr —
R
— u)2 J mray'dr=0.
(6.33)
The variable cp here represents the total angle of rotation of the blade
relative to the feathering hinge as a consequence of deformations of the control
and as a result of the kinematic action of the flapping compensator.
The solution to this system of equations, just as for the system (6.10),
can be obtained by means of B.G.Galerkin's method, if we put
J
9 = 90.
where cpo is a function only of time and does not depend on the blade radius.
Let us write out the computational formulas for the case where the vibra-
tion mode in the flapping plane is represented by means of only the zero r and
the first y harmonics of the natural blade vibrations. In this case, the coef-
ficients of the matrices entering the equation of the forai of eq.(6.l6) can /409
be determined by the following expressions:
a) Inertia coefficient matrix C:
C=
'U
■-21
'-I2
'-22
C32
'^33
]■
(6.34)
A42
where
R
<^i2= — - — ( murdr;
R
^21= — - — \ma rdr;
'ah J
^22=1;
R
<^23= —. — {tnoydr;
fahJ
R
^32= ——\maydr;
^33=1;
b) Damping coefficient matrix D:
vfhere
D-
rf,l rfi2 ^13
"21 "22 ^23
"31 "32 "33 J
d,z=\cl~^\br^.ydr-
2 ^A.A J
"22 — •■
'a.*
+rf/r;
(6.35)
(6.36)
(6.37)
ZWD
443
d3i = YCl^^br^ydr;
I
R
c) Stiffness matrijc A:
A =
(221 ^22 ^23
0a
where
33 J
^»f '
«22=/il =
«23='''Po;>,^;
a33=<;
d) Centrifugal and aerodynamic stiffness matrix B:
B =
6,1 6,2
"21 ''22 "23
*32 *33 J
where
*u = i;
6,2= ^c" -^
2 */,.,
69, = \ ma rdr;
*22=1 + -^^!-,-^
2 «'/,
*23 =
»
= — ■ I mar'idr;
R R
[br-^ydr^^-^W {bydr
h — — JL/." P
C/09 C- —
^^ 2 " /,
(6.38)
(6.39)
(6.40)
(6.41)
444
II
PM'"^"'-
b^=k-
The characteristic equations for this case will have the same form as in All
the preceding case [see eq.(6.28)]:
Here,
where
•^0 — ^ii'->o~r'-i2' o>
^2=^U'->3T'^12' 3I
D^ = b,,S^^dnS^ + CnST\-d,^R^-\-by^T^-^d,^T^-^c,J-^;
^2 = bnS, + ^,,57 + ^^13^7 + bnT, + ^,2^;
^3 = '^ll'^8 + '^12^8;
/=', = 6„56 + &,27'6;
•^2 = ^11'^7 4~^12' 7!
•-*0 ^ ^22^33 '•23''32>
O 1 = C22W33 "T ^33"22 ^23^32 '^32"23>
Sj = &22<^33 4" ^33^22 "I" ^22^33 — <^23"32 "23*^32 "32C23>
•J 3 = '^22'-33 "T '^33^22 ^23^32'
•^4 = ^22*^33 "f" ^33^22 — ^23^32 ~" £'32"23l
"J 5 = <l22'*33 "T ^^33" 22 023'^32>
•Sg = b^ib^s ''23''32>
S-j = ffl22''33 "F ^33^22 '^23''32>
08 = fl22^33>
"o^^^2l'^32>
"l = ^21^*32 "T" ^32"21 — <-22"31'
^2^''2l'^32 I ^32^21 "r"2l"32 ~~ "22"3i;
(6.42)
(6.43)
(6.44)
445
f\3 = 0,21^32>
/(4= O21W32 T''32'*21 ''22"31»
/<5 = <Z2t^32 — ^^22" 31'
°6^^^ ^21^32'
/<7^ 021^32!
'0^^ ^21^331
/ 1 ^ CjaOai — C21U23 — C33«21 >
/ 2 = U2^2l — ^21^33 ^33^21 — '*2l'*33>
73= — 021^^33 — ^33*^21;
'4 ^= ^23^31, — ^21^*33 — "3Sp2l !
/ g = ^23^31 ~~ '^2l'^33 — '^33"'2l ;
^6^^ ^21^33!
y 7= — '^21 ^33 — '^33^21;
/8= 021^33*
Zog
The values of the critical angular
mined by simultaneous solution of two e
30 Xc^ %
Fig. 4. 27 Critical Flutter Rpm as
a Function of Blade Balancing, for
Two Values of its Flexural Rigidity.
velocities of a given case are deter-
quations obtained from eq.(6.42) if we
set X = ip, as in the case of the blade
elastic in torsion.
7. Calculation Results
To illustrate the effect of
flexural rigidity of a blade, Fig. 4. 27
gives the critical flutter rpm as a
function of 5Eo, ^ for a blade of mass
constant over its length and with
balancing. The curves are plotted for
two values of flexural rigidity of the
blade. The degree of rigidity is
characterized by the values of the fre-
quency of natural bending vibrations
of the flxst harmonic of a nonrotating
blade poj^ . The cases investigated are
those of blades with the usual magni-
tude of flexural rigidity, at
POi/Ptw ~ 0.3 (solid curve) and of
Poi/Ptw ~ 3«0 which corresponds to a
very rigid blade (broken curve) .
The share of bending in the mode of blade vibrations during flutter can /413
be estimated from the ratios 6j/6g plotted for a mmber of points on the same
graph. The quantity 61 /6p is equal to the ratio of the blade tip deflection in
bending relative to the shape of the first harmonic to the displacement of the
446
tip during vibration of the blade as a solid body ( shape of the zero harmonic) .
It follows from these data that for a blade with constant mass and balancing
over its length, consideration of flexural deformations with respect to the
first harmonic does not greatly refine the calculation results.
8. Bending Flutter
The results presented above cannot be extended to all designs of rotor
blades. In individual cases, vibrations with primarily bending of the blade
occur during flutter. This type of flutter is usually called "bending flutter".
In bending flutter, the blade vibrates in the flapping plane with a mode
close to some harmonic of the natural -vLbration of the blade in bending and is
twisted with respect to a mode close to that of the first harmonic of the natural
vibrations in torsion. As already noted, flutter with modes of subsequent har-
monics of natural vibrations of the blade in torsion is theoretically also pos-
sible. However, the critical rpm of such flutter is several times greater than
the maximum rotor rpm.
The previously examined flapping
flutter can be regarded as a particular
case of bending flutter in which the
blade vibrates with a mode close to that
of the zero harmonic of natural vibra-
tions of the blade in the flapping
plane .
p
y
X
y
Ps
^
y
too
/
/
■».
— ■
—
\
60 (O
Fig. 4. 28 Variation of the Real and
Imaginary Parts of the Roots of the
Characteristic Equation as a Func-
tion of Rotor Rpm.
For footnote see next page.
To each harmonic of bending vibra-
tions of the blade there corresponds a
separate flutter zone in which the vi-
brations are characterized by specific
parameters inherent only to this zone.
Blade vibrations with different modes of
flutter may occur quite independently.
The mode of flutter having the lowest
critical rpm is practically the first
to be detected. Most often, this form
is the flapping mode of flutter. How-
ever, we can mention a number of par-
ticular cases in which the critical rpm
of some bending mode of flutter proved
to be below the critical rpm of the
flapping mode.
As an example, let us discuss
flutter of a blade with tip loading.
This case is of practical interest for
jet helicopters with an engine installed
at the blade tip*.
hhl
Figure 4»28 shows the change of the real and imaginary part of the roots
of the characteristic equation (6.42) with respect to rotor rpm. The roots of
the characteristic equation (6.42) were calculated for a blade with a tip load-
ing approximately equal to the weight of the blade itself.
Figure 4.28 indicates that, in this case, there are two flutter zones; the
flutter zone appearing first relative to rotor rpm is distinguished by a vibra-
tion mode having a high content of blade bending. Therefore, this zone is usual-
ly called the zone of bending flutter.
It is possible to trace the manner in which the zone with the vibration /414
mode having an increasing share of bending with increasing tip loading begins to
separate from the zone of flapping flutter as the blade-tip loading gradually
increases. At certain loading, these zones may separate into two different
flutter zones.
Figure 4.29 shows the flutter zone at a relatively small tip loading, equal
approximately to 1/5 of the blade weight. In this case, the characteristic form
of the zone of flapping flutter is distorted and the second zone begins to sepa-
rate from it.
equal to 42^ of the blade weight f = 0.42] and approximately equal to the
Figures 4*30 and 4*31 show the flutter zones for a blade with a tip loading
blade weight ( = l.l). In the latter case, the flutter zone separates into
two different zones of flapping and bending flutter.
6,
Figures 4.29, 4*30, and 4«31 give the values of 6 = -^ — characterizing
the vibration mode on flutter and the quantities p representing the ratios of
flutter frequency to rotor rpm:
tflu
It is of interest that the share of the flapping mode of vibration in /4l6
the bending flutter remains rather large in all cases, whereas the share of
bending in the flapping flutter may be almost completely absent in certain cases.
It should be emphasized that, for blades with tip loading, the critical
rpm of bending flutter is appreciably below the critical rpm of flapping flutter,
and that there is a weak dependence of critical rpm on the blade balancing.
This fact greatly complicates the problem of developing blades for jet heli-
copters.
* The results of the calculations given here (in Subsects.7 and 8) were obtained
by. V.M.Pchelkin.
448
/415
W i/szc
Fig. 4- 29 Flutter Zones with Blade Tip Loading Referred to
n.
Blade Weight ^° =0.20
w t/sac
Fig. 4*30 Flutter Zones vd.th Blade Tip Loading Referred to
Gio _
Blade Weight
= 0.42.
449
Vt/SiC
eo
w
20
F-r.i8 S=ij S=t.i9 s=v S=ws 6=-ijoi
p^it.S p = 5 p=S.f5 p=SJ3 p=S.5l p=S.57
20
22
2»
2B
28
30 X,^ %
Fig. 4.31 Flutter Zones vrith Blade Tip Loading Referred to
Blade Weight -^^^^— = 1.1.
9. Approximate Method of Determining the Mode of
Bending Vibrations in Flutter
If, in the first equation of the system (6.33), we discard terms of blade
vibration damping as well as the small term mocp, then we can write this equation
in the form
my+\EIy"r-{Ny'\~clQbU\. ■ (6.45)
Setting approximately U = our, we can represent the solution in the form
, } (6.46)
?=sin^^/,
y=y^smp^^^
where pf^u is the vibration frequency of flutter.
The calculations of bending flutter show that the frequency pf^^ can be
approximately set equal to the^frequency of natural vibration of the blade in
torsion and twist, i.e., Pfiu = Ptw*
We assume that
y^rT
J
ML
450
where
6j = coefficients of deformations;
y^^' = mode of .the j-th harmonic of the natiiral blade vibration.
Substituting eq.(6.47) into eq.(6.45) and applying B .G .Galerkin' s method,
we obtain expressions for determining the coefficients of deformation 6^ :
»/ _ y;^/
(6.Z^)
Here, V " / \ " /
^, = JVVW; (6.49)
^^= /n/? ' (6.50)
where
is called the equivalent mass of the blade during its Ad.bration relative to the
shape of the j-th harmonic.
Yj "^ mass characteristic of the blade during vibration relative to the
same harmonic;
Pj = frequency of the j-th harmonic of natural vibration of the blade in
bending; for the zero harmonic y , we can set pj = ou.
It follows from eq.(6.48) that the share of one or another harmonic of
bending vibrations in the mode of flutter depends primarily on two parameters:
relation of flutter frequency and frequency of natural vibration of the corre-
sponding harmonic, and magnitude of the integral Aj .
For example, the share of the flapping vibration mode ( j = O) in the mode
of flutter is smaller, the higher the "harmonic" of flutter, i.e., the ratio of
flutter frequency to rotor rpm. Here, we should note that since Aq is always
greater than zero, i.e..
Ag^^b~r^dr>0,
then the flapping mode will always be present in the vibration mode during
flutter. This conclusion is rather important and indicates, in particular, that
it would be incorrect to calculate flutter of some vibration mode without con-
sideration of the flapping mode.
The content of the first-harmonic natural vibration mode in bending in-
creases as the frequencies of the natural vibrations in bending pi and the fre-
quency of flutter pfiu close to the frequency of natural blade vibrations in
451
torsion are approached.
However, in this case the magnitude of the integral A^ is very substantial:
1
The calculation of this integral shows that, for the vast majority of
blades, this integral is close to zero so that the content of the first -harmonic
natural vibration mode in the vibration mode with the frequency pn^ is quite /41B
small. This explains the relatively rare appearance of bending flutter.
A quite different picture arises when concentrated loads are mounted to the
blade tip. The node of the shape of the first harmonic in this case shifts
toward the blade tip and the absolute value of the integral A-^ begins to in-
crease. Correspondingly, this causes an increase in the content of the first
harmonic in the vibration mode with a frequency Pfm*
Having assumed approximately that the vibration mode during flutter can be
calculated in a form of eq.(6.47) where the coefficients are calculated by means
of eq.(6.4B), we can develop a simplified calculation method for bending flutter.
Section ?• General Method of Calculation of F lutter and
Bending Stresses in the Rotor Blade during Flight
1. Calculation Method and its Possibilities
All methods presented above for the calculation of flutter were based on
a number of assumptions which, in many cases, it would be desirable to discard.
These assumptions include the following:
1) In the calculation of aerodynamic forces, the nonlinear dependence of
the aerodynamic coefficients on the profile angle of attack was disregarded.
Consideration of this dependence may have a substantial effect on the critical
rpm and especially on the character of amplitude build-up of oscillations in
flutter.
2) In calculating the aerodynamic forces under conditions of forward flight,
the flow compressibility was accounted for by introducing only values of Cy and
Xf averaged with respect to the rotor azimuth. Under conditions of forward
flight these quantities periodically change with respect to rotor azimuth, which
may have a noticeable effect on the critical flutter rpm.
3) Consideration of the forces of friction in the feathering hinge, which
- as is known - have a strong effect on the critical flutter rpm, was quite
arbitrarily done, by linearization of these forces.
In this Section, we will derive a method for calculating the bending and
twisting (binary) blade vibrations of a helicopter in flight, which permits dis-
carding these assumptions. This method makes it possible to determine the
452
bending stresses acting on the blade in the absence of rotor flutter and at
stable blade vibrations. If flutter is possible in the operating regime of the
rotor under consideration, then calculation by this method permits determining
the process of divergent blade vibrations and thus investigating the phenomenon
of flutter.
The calculation method is based on the approximate solution of differential
equations of blade vibration. In this case, B.G.Galerkin's method is used for
determining the form of blade deformations at some instant of time, while the
method of numerical integration of differential equations is applied for deter-
mining the overall process of blade motion with respect to time. B.G.Galerkin^s
method permits transforming the system of partial differential equations into a
system of ordinary differential equations and to use numerical integration for
solving this transformed system.
As applied to stress analysis, the method permits accounting for torsional
deformations of the blade in calculating the bending stresses in the flapping
plane. Under the effect of constant and variable external forces in flight, the
helicopter blade is twisted through some angle ?? which. is time-variant and /Z)l-9
differs with respect to blade length. Torsional deformations of the blade
change the angle of attack of its sections, which in turn leads to the genera-
tion of additional constant and variable aerodynamic forces. These auxiliary
forces must be taken into consideration when calculating the bending stresses of
the blade. If this is not done, good agreement between calculation and experi-
mental data is quite impossible.
When applied to flutter calculations, the proposed method is not too con-
venient in practical application, since it does not permit an exact numerical
determination of the parameters characterizing the limit of flutter. The flutter
limit can be established only in first approximation by visual inspection of
ciorves describing the blade motion for parameters close to this limit; similar-
ly, it is impossible to determine, with the required accuracy, the margins of
flutter based on parameters used in practice for evaluating the rotor from the
safety angle. The described method basically permits only a determination
whether or not flutter occurs in the flight regime under consideration and a de-
scription of its evolution.
Nevertheless, the method has a number of important advantages in comparison
with methods that use the roots of the characteristic equation and generally in-
vestigate flutter only in a linear array. It is difficult to imagine any other
method which would permit such a complete and accurate consideration of all non-
linear dependences, both in the magnitudes of aerodynamic forces and in deter-
mining friction forces, as is offered by this method in combination with numeri-
cal integration of the equations with respect to time. Consideration of these
dependences is highly important for flutter calculations. Therefore, it is
preferably used in control tests and check calculations, after determining the
flutter parameters by means of the roots of the characteristic equation.
Of great importance lor practical use is the fact that this method, without
excessive complication of the calculation, permits considering the elastic
couple between blades through the automatic pitch control, even at different
rigidity of the longitudinal and lateral controls. Without consideration of
453
this couple, a calculation of torsional deformations of the blade cannot lay
claim to accuracy.
2. Basic Assumpti ons and Suggestions
To derive the differential equations of motion of the blade, let us examine
the conventional type of rotor with individual hinge attachment of each blade
to the hub and vrLth control through the swashplate. In determining the angles
of twist of the blades as a consequence of deformation of the control system,
we will consider that the rigidity of the longitudinal and lateral control loops
differ. We will consider deformations of all control loops of both cyclic and
collective pitch control including deformation of the swashplate, which is
necessary when external forces generated by the rotor blades are locked on the
plate.
The motion of an individual rotor blade will be considered to consist of
flapping and bending vibrations in the thrust plane and of torsional vibrations,
both due to deformation of the blade and of the control system and to the kine-
matic action of the swashplate and flapping compensator. As above, we will dis-
regard blade vibrations in the plane of rotation.
With respect to blade design, let us use the following stipulations: Let
us consider that the flexural axis of the blade is rectilinear and coincides
with the feathering hinge axis. The plane of least rigidity of the blade will
be assumed to coincide with the flapping plane, i.e., with the plane going
through the axis of rotation of the rotor and perpendicular to the axis of the
flapping hinge. The flexural deformations of the blade will be determined in /420
this plane.
The rotor blade will be considered as a beam with the parameters continu-
ously distributed over its length.
3. Differential Equations
With consideration of the above stipulations, the differential equations of
blade vibration can be written in the following form:
where
y = displacement of points of the elastic axis of the blade relative
to the plane of rotation of the rotors;
9 = angle between the profile chord and plane of rotation of the rotor;
m = mass of the blade per unit length;
I„ = moment of inertia of the blade per unit length relative to its
flexural axis;
EI = flexural rigidity of the blade;
GTtw = torsional rigidity or twist of the blade;
454
a - distaaice from the flexural axis of the blade to the centers of
gravity of its elements, with the shift of the e.g. toward the trail-
ing edge of the blade considered as positive;
u) = angular velocity of rotation of the rotor;
r = distance from the axis of rotation to the examined blade element;
N = centrifugal force in the blade section:
N=(iy^^ mrdr;
T = aerodynamic load per unit length in the flapping plane;
Sitae r ~ aerodynamic torque per unit length relative to the flexural axis.
The method of determining the aerodynamic loads will be described in Sub-
section 6.
The dots in eqs.(7.l) denote differentiation with. respect to time and the
primes, with respect to the blade radius. In differentiating the function 9
with respect to the radius we should not introduce the geometric twist of the
blade into the value of cd', assuming that cp' = ??' where ■& is the elastic angle
of twist of the blade.
4. Boundary Condit ions of the Problem
For the type of rotors discussed here, the boundary conditions in the blade
root can be written in the form
(7.2)
N
where
Mo = bending moment in the blade root;
Mt = external torque relative to the feathering hinge axis due to
forces acting on the blade, with the pitching moment considered
as positive;
K = flapping compensator;
mi, = mass of the helicopter without blades; /421
Z/[EIy" ]o = sum of forces striking the helicopter hub from all rotor blades
^ (the index N denotes the blade numeral) ;
Mfj = moment of friction acting on the blade in the feathering hinge
from the side of the rotor hub, with the pitching moment con-
sidered as positive;
c^q = equivalent rigidity of the control system reduced to the axial
hinge of the hub (the method of determining this rigidity will
be given in Subsect.5);
Y = angle of rotation of the blade root in the axial hinge of the
hub, as a consequence of deformations of the control system.
455
In deriving these boundary conditions, friction was taken into account only
in the axial hinge of the hub loaded by centrifugal forces. Usually, we can
disregard friction in the other hinges of the hub and of the control system.
With a sufficient degree of accuracy, the second boundary condition of
eq.(7«2) can be replaced by the condition
j/o = 0. (7.3)
5. Determination of Equivalent Rigidity of the Cont rol System
To use the third boundary condition, we must determine the magnitude of the
equivalent rigidity of the control system c^g . This value can be determined if
the angles of twist Yn °f ^•ll ^b blades of the rotor in the axial hinge of the
hub are known.
The angle of rotation of the N-th blade of the rotor Yn is related with the
deformations of the individual control loops by formulas derived previously [see
eq.(4.9)]:
Y^ = Y..,+Y.sin^<,^' + Y,cos4,L^' + Yi:!',
(7.4)
where N = 0, 1, 2, 3, •••» '^■b ~ !•
Solution of the system (7-4) yields the following expressions for its un-
known Yep, Yx, Yz, andYi'^p:
^'■P ' z, ^j'^'^'
Y^ =
-J-^^N^^^f^l
* N
)iN).,
Yw - Y..n - Y^ sin <)-(,^) - Y^ cos<]>(,^)
a.p
ta.p ■
{1.5)
The magnitude of the hinge moment acting on the blade from the control can
be expressed in terms of rigidity and deformations of the corresponding control
cables
Moon =% Yep +^.Y.sini.i^> + c,Y.cos^l.^;+ c,.^y1C?.
(7.6)
where c<s.p, c^, Cj, , and c^^p are the rigidities of the collective pitch control,
lateral and longitudinal controls, and swashplate, respectively, reduced to the
axial hinge of the blade.
If we represent the magnitude of the hinge moment due to the control in /422
456
the form
JVicon —^tq 'A^> f 7 7^
then the equivalent control rigidity can be deterniined by the formiila
<'=<^..y, f.;,+c,f,sintL^'+r,YzC0s4.i^> + c^^Y'j;>, (7-8)
where the vinculimi denotes that the given magnitude of twist pertains to the
value Yn •
6. Determination of Aerodynamic Forces
To solve a system of differential equations (y.l), it is necessary to de-
termine the aerodynamic forces and torque entering the equation.
It is known that, during flow past the blade profile in flight, the angles
of attack of its sections may vary within wide limits, even to the extent that -
on the retreating blade - the flow passes over its root parts from the side of
the trailing edge. Flow-separation conditions occur at the blade tip in certain
regimes. At high flying speeds and at appreciable peripheral rate of rotation
of the rotor, the effect of flow compressibility has a considerable influence on
the magnitude of the aerodynamic forces. Therefore, a determination of aerody-
namic forces acting on the helicopter blade should take into account the non-
linear dependence of the aerodynamic coefficients on the angle of attack a and
the Mach number. Correspondingly, the expressions for determining the aerody-
namic forces should be written with consideration of the possibility of a wide
change in the angles of attack. At the same time, we can make use of the gener-
ally employed assumption of smallness of the displacements y and angles of rota-
tion of the blade sections cp. Therefore, to determine the aerodynamic forces
the following expressions can be used:
(7.9)
where
Cy and Cx = aerodynamic lift and drag coefficients;
m^ = torque coefficient of the profile, with Cy, c^ , and m^ deter-
mined from the results of downwash exposure as a function of
the section angle of attack a and M;
p = air density;
b = blade chord in the examined section;
Xo = distance from the leading edge to the flexural axis of the
blade;
Ux and Uy = mutually perpendicular relative velocity components of the
flow in a plane normal to the elastic axis of the blades,
with Ux being parallel to the plane of rotation of the rotor
and Uy perpendicular to Ujt ;
457
I ■ III I 11 II II
U = total magnitude of the relative velocity of flow past a profile in
a plane normal to the elastic ax±s of the blade.
The magnitude of the relative velocity U can be determined in terms of /423
its components
where
Here,
6^^=o)r+Vcosa^sirnj)^; (7.II)
Uy=wRl-y~V cosa^cos%?. (7.12)
cbR = tip speed of the blade;
V = flying speed of the helicopter;
or^ = angle of attack of the helicopter rotor in the shaft axes, i.e.,
angle between direction of flight and plane of rotation of the
rotor;
lift = azimuth angle of the blade;
3 = y' = angle of inclination of the elastic axis of the blade in the flap-
ping plane;
X = velocity of flow through the rotor referred to the peripheral
blade tip speed ouR, with the direction of X coinciding with the
axis of the rotor shaft; when the flow passes through the rotor
from the bottom up, X is considered positive.
The relative velocity of flow is determined by the formula
X=(x/^a^ + ^.^^, (7.13)
where
V cos Ol
Here, Vj„a is the induced part of the velocity of flow, also referred to cuR.
The induced velocity Vmd is a variable with respect to the rotor disk area
and to time.
In a number of flight regimes, the variable part of the induced velocity
increases so much as to lead to the occurrence of appreciable variable stresses
in the blade (see Sect .8, Chapt.I of Vol.Il). To determine stresses in the
blade with consideration of the variable field of induced velocities, it is sug-
gested to use the calculation method which involves calculation of the induced
velocities. If we limit ourselves to a consideration of only the constant com-
ponent of the induced velocity, then its value can be determined from the formula
Ct
4 y^^i^+OcoJ^'
458
(7.14)
where
Here,
voay and Xp^y = components of the induced velocity and flow-through ve-
locity, constant with respect to the azimuth and average
with respect to the radius of the blade;
Cj = thrust coefficient of the rotor:
C = ^/v»^
Trot ^ rotor thrust;
F = rotor area.
The angle of attack of the blade sections, needed to determine the aerody-
namic coefficients, can be calculated as
where
cp = angle of setting of the blade profile;
f = angle of inflow: /424
<^^fan-'~. (7.17)
'-'X
The angle of setting cp is a variable with respect to both blade radius and
time. It consists of two parts:
where
9=T,+», (7.18)
1] = angle of rotation of the blade in the feathering hinge as a conse-
quence of the kinematic action of the swashplate and the flapping
compensator, including also the geometric twist of the blades;
?? = angle of elastic twist of the blade, with the angle j? determined by
solving the system of differential equations (7.I).
The angle 11 is determined by the expression:
Tl=9o + Av<,„-e,sin^^-92COS<l>^-rjJo. (7.19)
Here,
00 = angle of setting of some blade section taken as the point of
reference at 3o = 0; this angle is usually called the "indi-
cator" angle of setting since its value is often given on the
instrument panel of the pilot;
^gooB ~ geometric twist of the blade;
9i and 02 = angles of cyclic pitch control caused by tilting of the swash-
plate;
Po ~ angle of rotation of the blade in the flapping hinge.
The Mach niimber, also needed for determining the aerodynamic coefficients,
459
I.
1
is calculated by the formula
M =
a*
where a^^ is "the speed of sound.
^=— ■ (7.20)
Thus, eqs.(7«l) together with eq.(7«9) make up a set of partial differential
equations with coefficients representing complex nonlinear functions of vari-
ables .
7. Method of Solving the Differential Equations
The method of solving eqs.(7«l) most convenient for practical use at the
present state of the art of computer technology is the method of numerical inte-
gration of the equations of blade motion with respect to time, in which the
blade deformations are determined by B. G . Gal er kin's method. In the formulation
of the problem adopted here, this method permits obtaining the most accvirate
results.
In determining the bending strain of a blade, it is natural to represent
the solution by means of functions which are natural vibration modes of the
hinged blade in vacuum. The peculiarities in the distribution of rigidity and
mass characteristics over the blade length and the boundary conditions of the
problem have already been covered by such functions. We set
^=2)w^ (7.21)
where
y •'' = mode of the j-th harmonic of natural blade bending vibrations;
6j = coefficients of blade deformation with respect to the j-th
harmonic.
In determining the torsional strain, certain difficulties are produced /42$
by the fact that the deformations of the' controls vary substantially, depending
on the direction of the moment of friction in the axial feathering hinge and on
forces generated at the swashplate by the totality of rotor blades. The relations
between the twist of the blade root and of all its longitudinal sections also
vary, depending on the conditions of the effect of these factors. To take this
into account, we must introduce some additional variable into the calculation.
Let us study this problem in greater detail. To determine torsional de-
formations by the Galerkin method, just as in determining bending deformations,
it is logical to use the modes of natural torsional vibrations of the blade in
vacuum. Here we can use various systems of eigenf unctions, differing by the
boundary conditions in the attachment of the blade at the root.
The solution to eqs.(7.l) is simplest if we assign the blade twist by means
of natural torsional vibration modes, determined for a blade represented as a
beam with a fixed value of torsional stiffness at the point of attachment
(Fig, 4. 32a). This method of solution is quite common in practice. However,
460
here the problem basically reduces to a calculation of the vibrations of an iso-
lated blade, since the use of the indicated modes precludes the possibility of
accounting for the elastic couple between the blades through the swashplate.
The effect of the moment of friction in the axial hinge of the hub cannot be
fully covered. Actually, the elastic tvd.st of the blade root is determined by
the magnitudes of the moments M^oon acting on the control system; furthermore,
the magnitude of these moments at known moments due to the blade M^, depends on
the direction and magnitude of the moment of friction:
AI<^j=A^(^)+yw<^). (7.22)
Therefore, blade twist at the root, and consequently the connection between the
twist of all sections of the blade length, are related with the magnitude of the
moment of friction. This effect cannot be accounted for if the indicated con-
nection between the twists is fixed by vibration modes used in the calculation.
It follows from the foregoing that this calculation method should be con-
sidered invalid as applied to real helicopters. It can be used only in indi-
vidual - rarely encountered - particular cases.
To take into account the couple between blades through the swashplate and
the effect of the moment of friction in the feathering hinge, we could use a
system of functions representing the modes of natural torsional vibrations of
the blade in the form of a free beam unattached at the root (see Fig.4-33D).
However, owing to the discrepancy of boundary conditions, the use of such func-
tions might lead to a solution of only an approximate type. Actually, the modes
of torsional deformations thus obtained will substantially differ from the real
modes. This difference will be especially pronounced in twist of the root por-
tions of the blade where, for a free beam, the torque diagram drops to zero.
All these considerations necessitate applying a nonorthogonal system of
functions to this problem, as shown in Fig. 4.32c. In this case, the twist of
the blade can be represented in the form
?=t1 + Yo+2y,»'*'. (7.23)
where k = 1, 2, ... .
Here,
Yo = angle of twist of the blade as a consequence of deformation of /426
the control system;
^(k) = mutually orthogonal modes of natural torsional vibrations of a
blade rigidly fixed at the root;
Yt„= unknown coefficients of the torsional deformations of the blade.
Thus, the blade twist is represented by a system of orthogonal functions
3?^"^ , supplemented by a function #°^ =1 nonorthogonal to this system.
Equation (7.23) can be written in the form
461
f = n + ^y^bW,
(7.24)
where k = 0, 1, 2.
This form of representing the blade twist creates certain complications in
the calculation, produced by the nonorthogonality of the functions d^ . Never-
theless, we must put up with these complications in order to accoiint for all of
the above highly important factors.
Beam vith elastic
attachment
Beam with rigid
attachment
Fig. 4.32 Modes of Natural Torsional Vibrations of a
Beam with Various Attachments.
8. Transformation of Partial Differential Equations
into Ordinary Differential Equations
Having represented the solution of system of differential equations (7«l)
in the form of eqs.(7.2l) and (7.24), let us apply the Galerkin method. For
this, let us twice differentiate eqs.(7.2l) and (7.24) and substitute them, to-
gether with their second derivatives, into eqs.(7.l).
form:
The second derivatives from eqs.(7.2l) and (7.24) will have the following
(7.25)
462
TABLE 4.1
j =
7 = 1
;--2
y-3
* =
k^\
*^2
y =
y-i
;-2
y-=3
*-0
k=l
k=^2
h
81
Bj
53
Yo ■
Yi
Y2
»0
»i
5j
83
Yo
Yi
Y2
/ =
*oo
*0I
*02
*03
.Coo
coi
f02
«00
«Q1
''02
"03
*00
*01
*01
-^0
/ = 1
*I0
ftn
*I2
*I3
ClO
'u
C12
"10
"11
"12
"13
*10
*11
*12
=-4.
1 = 2
*20
*21
*22
*23
^20
C21
<^22
"20
"21
"^22
"23
*20
*21
*22
=^2
/ = 3
*30
*31
*32
*33
C30
C31
er,2
«30
"31
"32
"33
*30
*3.
*32
=^3
* =
foo
«01
C02
«03
/o
/i
/2
*00
*01
*02
*03
'■>%+Clf
<o2/,
a.2/2
= So + Afjr
* = 1
"lO
cu
«I2
fl3
/l
ii
*10
*n
*12
*13
c«2/,
v?^.
=Bi
*=2
C20
Cj,
''22
^23
^2
L,
*2I)
*21
*22
*23
...2/2
v^i2
=B,
^
UJ
4>
We then multiply the first equation of the system (V.l) by y^"*^ and the /428
second by d^^^ and integrate all terms with respect to the blade radius. The
boundary conditions (7«2) should be accounted for in the integration. This op-
eration transforms the system of partial differential equations into a system of
ordinary differential equations relative to the new variables 6j and Ytw
For practical purposes, it is highly important what number of variables 6j
and Ytw^^ used in the calculation. Experience has shown that a sirfficiently
complete answer can be obtained if the bending strains are represented by means
of the first four harmonics of the natxoral blade vibration and the torsional
strains by two or - in the extreme case - by three harmonics. Thus, the problem
of bending and twisting vibrations of a helicopter blade can be solved with the
use, in any case, of seven independent variables. We will restrict the further
calculation to this number of variables.
The system of ordinary differential equations obtained from application of
the Galerkin method is written out in the form of a table (see Table. 4.1) •
All equations of this system represent the sum of the products of certain
constant coefficients and the unknown functions 6j and Ytw ^■'^d their second
derivatives. In Table 4.1, the coefficients pertaining to one equation occupy
one row. The known constants that do not change during the calculation are
written out in the squares of the table.
The independent variables 6j and Ytw ^^nd their second derivatives, entering
simultaneously all equations of the system, are extended with respect to the
vertical in a special row in the upper part of Table 4»1. The right-hand sides
of the equations are extended in a special column next to the table of constants.
The coefficients of the left-hand side of the equations of the system (see
Table 4.1) are determined after calculating the modes and frequencies of the
natural blade vibrations in bending and torsion. As stated above, in calculating
the torsional frequencies a blade rigidly fixed at the root is used.
A number of coefficients are determined directly during this calculation.
This concerns primarily the frequencies of the natiu^al vibration of a rotating
blade in bending pj and in torsion \^ , and also the coefficients into which the
mass characteristics of the blade enter:
ft
(7.26)
After calculating the modes and frequencies, we determine the coefficients
into which simultaneously enter the data obtained from calculating the blade
in bending and in torsion. These are the following coefficients:
464
A,^ =f my^'^y^ndr +x2pu)pa)/„_i_ ^pu) J moyU)clr +
+ ;cp(/)jmai,(Orfr.
a„^p) J mi/<'Vy)rfr+x2«.2%op^(/)/^+
_|_i
L
(7.27)
7429
The second, third, and fourth terms in these expressions are small in com-
parison with the first and can be neglected. At i 71^ j, the first terms of
eqs.(7.27) vanish by virtue of the orthogonality of the functions y^ ' ^ and j t
and the coefficients kij and a^j can be assumed as approximately equal to zero.
x.e..
Next, the coupling coefficients are determined whose value depends mainly
on the blade balancing.
At j = 1, we have
R I
(7.28)
The terms on the right-hand side of the system of equations (see Table 4.1)
are determined by means of the following expressions:
4-(B2(6jSlntl.-|-G2COS<;.) fmoi/Orfr— ? morpo dr ,
■J
5.= aJ?^r»<*'rfr-«)2/L
J/, — 1 JJiatr
(7.29)
465
p
Here, I^ = Golk + J I.Acpg,,,.^^ ''^ dr.
The first terms of eqs.(7.29) are those determining the value of the coeffi-
cients Aj and B^ . The following terms are small and can be neglected.
9. Determination of the Magnitude of the Moment of Fri ctlonin the
Feathering Hinge of the Hub
During the numerical integration of the equations (Table 4«l)» the magnitude
of the moment of friction can be obtained from the values of the torsional de-
formations of the blade determining the external torque in the feathering hinge
and from the direction of blade rotation in this hinge. In so doing, the magni-
tude of the moment of friction should be determined by a different method, de-
pending on which is greater in absolute value: the external torque in the
feathering hinge M^i or the maximum possible moment of friction Mf^ •
The external torque in the feathering hinge is determined by the formula /430
M,.= M^-M,„„=^ V,„vM<*>, (7.30)
where k = 0, 1, 2, 3, • • • •
Here,
Ml, = hinge moment due to forces acting on the blade;
Moon ^ ^eqYe "" moment relative to the feathering hinge due to the control
system; in conformity with this notation the pitching moment
due to the control is considered as positive just as in
eqs.(7.6), (7.7), and (7-22);
Vi^^'' = magnitude of the hinge moment in blade deformations with
respect to the mode of the k-th harmonic of natural vibrations
of the blade in torsion.
Modes of natural vibration normalized in some manner, for example, by the
quantity j?r ' = 1, will now be discussed.' Here, we asstmie that
The magnitude of the maximum possible moment of friction K't^^ is usually
determined experimentally in the laboratory. If the coefficient of friction in
the bearing f is known, then this magnitude can be determined by the formula
where
Np = centrifugal force acting on the bearing of the axial hinge;
r^e = radius of this bearing.
If JMhi I < JMfr^l, then Mf^ = -Mhi . In this case, the blade in the feather-
ing hinge does not turn, and cf^ = c^ =0. This condition permits determining
immediately '^o and Yo •
Zp66
If iMhi
} (7.31)
10. Sequence of Performing the Calculation
The system of differential equations (see Table 4.1) is written here in a
form such that its solution is conveniently found by numerical integration with
respect to time. During this integration, mainly the right-hand sides of the
equations will change. All coefficients on the left-hand side of the equations
remain unchanged during the calculation, with the exception of the coefficient
Cgq whose magnitude is recalculated at each integration step.
The numerical solution of the system (see Table 4«l) also represents the
basic part of the method of calculating binary blade vibrations presented here.
The calculation of blade vibration by this method is carried out in the
following sequence:
1) Calculate the modes and frequencies of natural blade vibrations in
vacuum. For calculation by this method, it is necessary to determine the first
four harmonics of flexural vibrations of the blade, including the so-called zero
harmonic of vibration of the blade as a solid body, and the first two harmonics
of the torsional vibrations of a blade rigidly fixed at the root. From re-
sultant vibration modes, determine the constant coefficients of the system of /431
differential equations (see Table 4«l)« In" the numerical integration of the
equations, all these coefficients remain unchanged with the exception of the co-
efficient Cgq whose determination is described in Subsection 5«
2) Select the parameters of the flight regime p , u), M-, ffh , 9© > 9i » ^s i^^
which the bending and twisting vibrations must be calculated.
Usually, these parameters are taken from an aerodynamic calculation of the
rotor and from calculation of the balancing characteristics of the helicopter.
However, another more natural method can be used. The calculation method pre-
sented here can be used as a method of aerodynamic calculation and calculation
of balancing, by adding a number of simple operations. The values of o-h and %q
can be obtained from the calculation if the values of thrust and propulsive
force of the rotor and the angles 0i and Gg necessary for fulfilling the flight
regime are prescribed and if the moments on the hub necessary for balancing of
the helicopter are determined.
3) At the initial instant of time, which is usually related with the azi-
muth angle tb ~ 0> assign arbitrary values of the variables and their first
derivatives 5j , YtH» ^j ^^^ YtM* To account for the coupling between the blades
through the swashplate, these values are assigned for all z,, blades of the rotor.
4) Determine the magnitudes of the aerodynamic forces necessary for calcula-
tion:
467
(7.32)
J
■where the value of T] is determined by differentiation of eq.(7.19)t
7]= — (1)9, cos ijj^ + 0)62 sin ij)^— xPp. (7*33)
Here, Po = ^ 6jp^^^ .
5) From eqs.(7.10), (7.11), and (7. 12), determine the velocity of flow past
the profile and its components, and derive the angles of attack of the sections
from eq.(7.l6). Use eq.(7«20) for determining the Mach number.
6) From the polars of the profile fed into the computer together with the
initial data, determine the values of c^, Cy, and m^. After this, making use of
eqs.(7«9), calculate the aerodynamic forces per unit length T and the torsional
moments 311^,^.
7) From the known values of T and ffiaer > determine the terms Aj and By
entering the right-hand side of the differential equations (see Table 4«l)»
8) To determine the value of c,^. it is necessary to know the values of
blade twist in the feathering hinge Yo ' ^°^ ^-^ ^b blades of the rotor. In
this case, c, , is determined by the method presented in Subsection 5.
9) Determine the value and sign of the moment of friction Mfp in the
feathering hinge (see Subsect.9).
After this, derive all coefficients of the equations (see Table 4«l) and
start with the solution.
10) The system of equations (see Table 4.1) permits determining all values
of 5j and Ytw i^ ^j » Ytw aJ^d the right-hand sides of the equations Aj and Bjj
are known at the azimuth ^^ in question. This fact permits its use in the /432
calculation program in the form of some operator of the type
h'\=p(h'\„,v- ^^-^^^
After applying this operator, determine the values 6^ and y'tw ^ 'the initial
instant of time.
11) The change to the next instant of time can be accomplished by means of
various methods of numerical integration of differential equations.
468
Good results are obtained by a system of formulas in which the transition
from the instant of time t to the time t + At is accomplished by two checks.
This system of formulas is illustrated for the example of determining the values
of the variable 6j . The index pertaining to the number of the harmonic is
omitted for simplicity.
First check:
Determination of 5.„:
Second check:
8<+4< = 8, + A^8,;
8/+4/ = /'(8)+4,, y\+M, <^t+iii).
*/ + ti/ + A(
(7.35)
8/+A< = 8<-l-A^8^v;
8<+4/ = /'(8/+i/, y'Va/, <i'<+A/).
The values of 6t+Atj S^+^t* ^t+At obtained as a result of recalculation are
considered final for the instant of time t + At .
Operations analogous to eq.(7.35) are performed on the coefficients of tor-
sional deformations. The change-over to the next instant of time is thus accu-
rately accomplished.
A simpler method of numerical integration can be proposed. This will be
presented in greater detail in Vol.11.
12) The type of problem investigated is important for the sequence of
calculation. If it is a question of determining the possibility of rotor flutter,
then the process of numerical integration must be carried out simultaneously for
all rotor blades and the value of 0,, must be determined at each instant of time.
The coupling between blades through the swashplate is taken into account by
calculating the quantity c^ ^ . If the question of investigating flutter is not
raised and only stresses in the blade are being determined, the problem is great-
ly simplified. In this case we can introduce into the calculation the assump-
tion that all blades of the rotor duplicate the motion of the blade in question,
and the process of numerical integration is performed for only one blade.
When determining c^ , in this case it is assumed that 7433
1*69
YMr)=Vo(C-^). (7.36)
where
i|r^°^ = azimuth angle of the blade with the ntunber N =
/ 9 H N whose motion is determined in the calculation;
Yo('l'b°^ z^j " coefficient of deformation of the blade with the
'' number N = 0, not at the azimuth %°^ in question but
at the azimuth i^i°'' g
YN^'b ) " coefficient of deformation of the blade with the
number N, when the blade with N = is at the azi-
muth ^i°' .
13) In determining the stresses, the nimierical integration is performed for
several rpm of the rotor until all values of 6^ and Ytw^'t 'two successively
calculated rpm differ less than the prescribed accuracy of calculation. This
will indicate that the process has converged. After this, the bending stresses
at each azimuth can be determined by the formula
=2v
(/■)
(7.37)
where o^^^ are the bending stresses of the blade with respect to a normed mode
of natural vibrations of the j-th harmonic.
Further reduction of the obtained data can be performed in any form, de-
pending on the purpose of the calculation. Usually, the amplitude of the
stresses is determined and the variation in stresses with respect to azimuth is
decomposed into harmonics.
14) In the investigation of flutter, the results can be evaluated after
studying the entire process of variation in the deformation coefficients during
several rotor rpm. This is not very convenient in practice since it requires
considerable graphic work for plotting the dependences 6j = f(i|f) and Ytw ^ f(t)«
Nevertheless, these drawbacks are compensated by the advantages of this calcula-
tion method.
The method presented here involves a large amount of work, but it is known
from practical experience in design shops that, if modern digital computers are
used, this method best meets the requirements in designing and perfecting blades
and permits introducing additional refinements into the results of the calcula-
tion based on an analysis of the roots of the characteristic equation.
Section 8. Experimental Investigations of Flutter
1. Ground Tests for Flutter
The features of helicopter design permit the performance of flutter analysis
470
of the rotor under safe conditions, with the helicopter on the ground. This
constitutes a distinct advantage of the helicopter over regular aircraft.
Ground tests for flutter are carried out for different purposes. Often
these purposes are purely of a research nature. In many cases, it is necessary
to check or refine - under full-scale conditions - the effect of various para-
meters on flutter characteristics, to evaluate the peculiarities of the develop-
ment and cessation of flutter and, finally, to simply refine individual moments
in the procedure of conducting such tests.
Nevertheless, in the overwhelming majority of cases these tests are /434
carried out for inspection purposes. Recently, it has become the rule that each
experimental helicopter must undergo flutter tests before the start of flight
tests. The actual margins to the onset of flutter are established in these
tests. If they prove to be too large, the designer can reduce them, for example,
by decreasing the weight of the counterbalance in the blade and thus lightening
it. In the case of insufficient margins, it is necessary to make some design
modifications and recheck them in tests.
The finally established flutter margins on an experimental helicopter will
later serve as criteria for evaluating the characteristics of other helicopters
of the same design in production at a series-production plant or in actual
service .
Usually, in developing a new helicopter it is possible to restrict the test-
ing to ground tests without the need for additional flight tests. In exceptional
cases in the past, it had been necessary to also conduct flight tests. As a
rule, there is no need for these.
Ground flutter tests are usually carried out in the following manner:
The helicopter is made fast on a special platform so that the possible oc-
currence of flutter and consequent failure of a part will not cause the heli-
copter to roll over. As is known, roll-over of a helicopter will cause the
blades to strike the ground and almost completely wreck the craft. In some
cases, there might be casualties. Generally, such does not happen in ground
tests for flutter, but the experimenter must always be prepared for any eventu-
ality.
To begin the tests, the rotor should be revved to the maxim.um rpm at which
flutter cannot yet occur. Then the rpm is gradually increased. Usually, this
increase is accomplished in steps of a certain quantity An, so that ng = ni +
+ An. Here ni is the initial value of the rpm and ng the new value. The quanti-
ty An is generally taken as about 2% of the operating rpm of the rotor.
At the new rpm ng, the rotor is held for some time (usually 1-2 min) so
that vibrations can proceed up to noticeable intensities; if flutter does not
occur, the rpm is again increased by the quantity -An until flutter does develop.
Flutter tests are usually greatly simplified if, to cause flutter, it is
not necessary to create initial disturbances as it is required in hard flutter
with an excitation threshold (see Sects. 3. 3 and 3«4). Therefore, an attempt
471
%
should be made in the tests to create conditions favorable for the occurrence of
soft flutter. Such conditions usually are present if a sufficiently large forced
motion is generated in the feathering hinge. For this, the control lever, and
along with it the swashplate, are deflected forward as far as possible. Usually,
this is limited by the fact that the blades begin to strike the supports of the
vertical overhang guard.
When a forced motion is created in the feathering hinge, flutter sets in
earlier with respect to the rotor rpm. Thus, pulling the control stick, in a
way, is a means of generating flutter. Here, the start of flutter tests is as
follows: The increase in rotor rpm by An is carried out at neutral position of/435
the swashplate, after vrfiich the control stick is pulled forward and the regime
is maintained with the stick deflected. If flutter does not occur, the control
stick is returned to the neutral position and the rotor rpm is again increased,
and so on, until flutter occurs.
Upon the appearance of flutter, if the oscillations build up rapidly, it is
first necessary to reduce the engine power sharply so as to cause a rapid drop
in rotor rpm. An additional means of stopping flutter is to return the control
stick to the neutral position.
In flutter tests, it is of great importance to achieve the maximum possible
rotor rpm. To prevent the rpm from being limited by the engine power, the rotor
is usually lightened meaning that the angle of blade setting is reduced. Experi-
ments have shown that the overall angle of blade setting has only a slight in-
fluence on the critical rpm of flutter and thus can be reduced without risk.
However, one definite limitation does exist. The lower the angle of rotor set-
ting, the sooner will the blade begin to strike the supports where the control
stick is deflected. Furthermore, severe lightening of the rotor is unwarranted
so that the maximimi rpm in the tests is limited not so much by the power as by
the mechanical strength of the engine. Therefore, the angle of blade setting is
selected as maximum in the tests but is kept at a value preventing the blades
from striking the supports when the control stick is deflected while maintaining
sufficient engine power for maximum possible rpm allowable for mechanical
strength reasons.
Flutter tests under ground conditions obviously are possible only if the
rotor characteristics are such that flutter will take place under these condi-
tions. On helicopters rated for service, flutter cannot occxor under ground con-
ditions. Therefore, to conduct ground tests for flutter, the rotor parameters
must be disturbed somehow. This is usually accomplished in the simplest way by
disturbing the blade balance, which can be achieved by attaching small weights
to the trailing edge of the blade. Occasionally, the balance is shifted by coat-
ing the surface of the blade close to the trailing edge with some kind of ma-
terial whose weight will shift the blade balance rearward. It is also possible
to introduce some elastic elements into the control loop. Thus, in conducting
flutter tests, the rotor parameters must first be changed so as to make occur-
rence of flutter possible.
When conducting the tests, it is necessary to provide for the recording of
various parameters to permit an accurate determination of critical rpm, fre-
quency, vibration mode, and deflection of the control stick at which flutter
began.
472
Without a determination of these parameters it is impossible to make a suf-
ficiently accurate evaluation of the flutter margin and to indicate what para-
meters should be changed to increase this margin.
When conducting the tests, the onset of flutter is detected by the pilot
from the disturbance of the blade coning angle and from the increase in fuselage
vibrations and, in the case of reversible control, also from vibrations of the
control stick. However, all these signs are sufficiently distinct only after
the vibration amplitudes reach extremely high values and conduction of the test
becomes dangerous. Consequently, it is desirable to stop flutter tests earlier,
before oscillations have time to develop. In this case, the pilot may easily /436
confuse flutter with the usually present distortion of the blade coning angle.
This is promoted by vibrations which, as a rule, arise in such tests owing to
wind and lack of controllability of the rotor.
In this case, the occurrence of flutter can be judged- only by recordings of
various factors that are characteristic for vibration. To determine the onset
of flutter and its parameters, the type of recordings made in the tests is of
great importance. It has been shown that it is not always easy to determine the
onset of flutter from a recording of the flapping motion of the blade in the hub
hinges, since flutter vibrations in these hinges lead only to a distortion of
the recording of the flapping motion caused by deflection of the control stick.
This is illustrated in Figs. 4-33 and 4*34 which show a recording of blade
motion in the flapping hinge (angle p), with the recording of weak flutter shown
in Pig. 4. 33 and of stronger flutter in Fig.4.34« As follows from Fig. 4. 33, a
determination of the onset of low-amplitude flutter would be difficult from a
recording of the angle p. The same is true with respect to recording the hinge
moment M^.
The onset of flutter is best reflected in the recording of forces in the
nonrotating control loops. The recording of forces in the longitudinal control
Piong is shown in the oscillograms (Figs. 4*33 and 4.34). It is easy to define
the onset of flutter from these recordings.
It should be mentioned that Figs. 4*33 and 4.34 show the recordings of anti-
phase flutter with an order m = 3 for a four-blade rotor. Consequently, the
frequency of the variable forces in the longitudinal control is governed by the
relation
P}o,g ^^Pjlu + f^fl"^ (8.1)
where
jOfi„^330 osc/m'in
Nfiu^ 184 fpm
When the control stick is deflected from the neutral position by even the
slightest amount Xp, the vibration frequency of the blade in flutter Pfju can no
longer be determined from recording the angle 3 (see Figs.4«33 and 4.34), but
can easily be calculated from eq.(8.l) since the value of the frequency Piong is
473
■p-
n~n3.rpm
' I I
Fig. 4.33 Oscillogram of Hinge Moment M^, Blade Flapping Angle P, Position
of Control Stick Xp, and Forces in Longitudinal Control Piong in
the Presence of Weak Flutter.
■Al'^V''! -vi- '" \Neutral position of stl
n'One rotor revo lut ion
\^ \j u \J \J' v; \} y T- J y y IX -'^v } V } • I n_ n'^ne roior revo lution
:^:lr-^~::zr^- ^ ' ! -J. ^^ J^y^U aX attA 7u&.-A7tKAi^AA rys^p;^^^^^^ at^^ttT'^^-A^-A- m a a a a L ;. ..1. . J . A . I —
^ ■ « *■ t^1»<ma^^ t t I 'l^f I * ff >
ri
-1-
. Onset of flutter.
I ! n-isirpm ,
! 1 I I I
n - rss rpm \
I I I I
,n =186 rpm
Fig. 4.34 Oscillogram of Hinge Moment M^, Blade Flapping Angle p. Position
of Control Stick Xp, and Forces in Longitudinal Control Piong i^^
the Presence of Stronger Flutter.
5
-<3
readily determined from the oscillograms.
If, after occiirrence of flutter, the control stick is returned to neutral,
the flapping motion caused by tilting of the swashplate will stop and the only
motion in the flapping hinge will be that due to flutter (see Fig.4«34)« In
this case, the frequency of flutter can be determined also from the recording
of p.
In the tests whose recordings are shown in Figs. 4*33 and 4*34, flutter was
caused by an increase in rotor rpm and by deflection of the control stick by an
amount Xp.
In the first case (see Fig. 4. 33), the rpm was raised to nfiu = 184 and, as
soon as weak flutter set in, it was stopped again by decreasing the rotor rpm.
The position of the control stick Xp had not been changed.
In the second case (see Fig. 4. 34), the rpm was raised somewhat more, up to
n = 186, causing stronger flutter to occur. At the start, the control stick was
retiorned to neutral without a change in rpm; this caused the increase in vi- /438
bration to stop, after which the rpm was lowered and the flutter disappeared.
It should be mentioned that the recordings shown in Fig. 4. 33 and 4.34 cor-
respond to rather weak flutter with a slowly increasing amplitude. Such flutter
is not always observed; often, the vibration amplitude increases much more
rapidly and the manipulation of the control stick, described above, becomes im-
possible.
As an example of such abruptly developing flutter. Fig. 4-35 shows an oscil-
logram of blade motion about the flapping hinge for another helicopter with a
three-blade rotor. To stop flutter on this helicopter it was necessary to reduce
the rpm as rapidly as possible.
The flutter vibration mode whose recording is shown in Fig. 4-35, is of the
in-phase type which means that the collective pitch control is loaded during the
vibration. This makes the recording of the swashplate slide vibrations, shown
in Fig. 4. 36, quite interesting. This recording was made with a CV-11 automatic
recorder.
The recordings shown in Figs. 4*33 - 4»36 are given only as an example and
in no way exhaust all possible types of flutter observed on helicopters. These
types may differ in modes of blade vibration, phase distribution of vibrations
over the blades (different values of m), frequencies, rate and character (soft
and hard flutter) of build-up of vibrations, and in numerous other features.
All these peculiarities must be taken into account in flutter tests and in pro-
cessing the obtained recordings.
2. Flutter Tests in Flight
Flutter tests in flight became necessary when it was found that, during
mass service of helicopters, there were individual cases of flutter in flight
when such flutter should not have been possible according to concepts held at
that time.
475
OS
10^ I
15-:?
ZItO
ri'ZIIO
2)10
One revolu tion of rotor
m--
ffwJ
/rmt
Fig. 4. 35 Oscillogram of Blade Flapping Motion during Violent Flutter.
Onstt of flutter
End of flutter
Fig. 4. 36 Recording of Forces in Collective Pitch
Control during Violent Flutter.
Tests were carried out, which showed that
the critical flutter rpm in flight is appreci-
ably lower than in ground tests.
The relation between the critical rpm in
flight and on the ground was calculated and
it became possible to define the characteris-
tics, checked in ground tests, that were
needed for prevention of flutter in flight.
The obtained conclusions can be used in de-
veloping new helicopter, making flutter tests
in flight for each type of helicopter wa-
necessary. It should be borne in mind that
tests with excitation of flutter in flight
are extremely dangerous. Such tests should
be performed only if absolutely necessary and
should be organized with maximum safety for
the crew.
Primarily, before starting the tests the
researchers should collect data ensuring that
abrupt development of flutter will not occur
in flight and that, if it does start, it can
be stopped again. Such data can be obtained
in cases in which unscheduled flutter sets in
during flight tests or diiring service on some
helicopter of the type in question. This
occasionally occurs as a consequence of some
operating error, for example, if the rotor is
revved to an rpm by far exceeding the permis-
sible maximum.
Ground tests can be used as an indirect
criterion for the degree of abruptness of
flutter. Experience has shown that the rate
of build-up of vibration on the ground and /440
in flight is determined to some extent by the
overall parameters. Therefore, in some cases
data of ground tests can be used as basis.
The only reliable measure for stopping
flutter in flight is a sharp reduction in
rpm. Therefore, to ensure definite stopping
of flutter it is necessary to have a large
rpm excess in a regime where flutter begins
in comparison with the minimum rpm at which
sible. During the tests, the pilot should induce flutter by raising
sharply reducing the rpm to the minimum possible for
bO
•H
H
C
•H
&
■P
bfl
13
bO
•H
ft
ft
cd
H
o
o
•H
o
m
O
P3H
flight is possxbie. uurxng
the rpm and stop flutter by
continuation of the flight.
All considerations referring to recording in ground tests hold also for
flight tests. However, we should point out one peciiliarity of vibrations during
477
flutter in flight, which distinguishes these vibrations from those observed in
ground tests.
In ground tests, forced flapping motion in the hinges caused by tilting of
the swashplate takes place almost exclusively at the frequency of the first har-
monic of the rotor rpm. In flight, the flapping motion contains also the second
and higher harmonics. Therefore, blade vibration in flapping flutter usually
generated at frequencies close to the second harmonic but generally not equal to
it, will lead to beats between the second harmonic of flapping and flutter vi-
bration. Therefore, flutter in flight is often perceived as beats.
As a typical example. Fig. 4.37 shows the recording of flutter in flight in
a regime where flapping in the axes of the shaft consists almost exclusively of
the second harmonic (see Fig.4«37b). This is explained by the fact that attach-
ment of the shaft was selected such that vibrations of the first harmonic are
eliminated in cruising flight.
The vibrations during flutter in this regime have well-defined beats (see
Fig. 4. 37a).
In all other cases, if the lower critical rpm is disregarded, flutter in /441
flight will not differ from that ob-
served on the ground.
Iflu fprnj Pflu "^'^In'ln
300
too
too
No flutter up to maximum
ro tor rpm, incl .
23W
21'/o
25%
26 fo
*W
Fig. 4. 38 Comparison of Experimental
and Calculated Values of Vibration
Frequency and Critical Flutter Rpm.
3. Comparison of Calculation and Experi-
ment under Conditions of Axial Flow
past the Rotor
In comparing calculation and ex-
periment, the initial rotor parameters
used in the calculation are of prime
importance, along with type of blade
balancing, rigidity of the control
system, and magnitude of friction in
the feathering hinge of the hub, as
well as reliability with which the lo-
cation of the profile focus is known.
Errors in determining the initial data
naturally affect the accuracy of deter-
mining the flutter parameters. There-
fore, in comparing calculation and ex-
periment it is desirable to eliminate
errors in determining the initial para-
meters. For this, the parameters
should be checked experimentally.
Balancing should be determined by weighing individual segments of the blade ob-
tained after cutting it.
To determine the control rigidity a special method of measuring dynamic
rigidity should be used, which will be taken up in greater detail in Section 6.
The use of other methods generally leads to misunderstandings and fallacies and
478
therefore should be discarded.
To check the position of the profile a.c. a segment of a full-scale blade
should be exposed to the air stream in a wind tunnel. In this case, it can be
expected that deviations in the aerodynamic characteristics due to design errors
of the blade profile and deformation in work will be refined to some extent.
Figure 4*38 gives the results of a comparison of calculation and experiment
for the Mi-4 helicopter. The solid curve shows the theoretically obtained de-
pendence of the critical flutter rpm on the effective blade balancing. The
circles mark the experimental results. Circle 1 with the forwardmost blade
balancing corresponds to the maximum rpm obtainable with a helicopter engine.
There was no flutter in this case. After attaching 0.46-kg weights to the blade
flaps, the experiment was repeated. There again was no flutter (circle 2).
Attachment of weights of 0.86 and 1.3 kg to the blade flap caused flutter
at rotor rpm of n = 18? and n = 173 respectively (squares 3 and 4 in Fig. 4. 38).
The frequency of blade vibration during flutter is indicated in the dia- /442
gram by squares 5 and 6, which should be compared with the theoretically deter-
mined frequency values shown by the dashed
curve.
After the experiments, the blades were
cut into segments and their effective
balancing was determined, which is noted on
the graph in Fig. 4. 38. The dynamic rigidity
of the control system was determined on the
same helicopter. The magnitude of friction
in the feathering hinge, which was highly
stable, was measured in the laboratory on
another hub of the same design.
These data indicate satisfactory (with
an accuracy to within 0.^% of the chord for
the value of effective balancing) agreement
of calculation and experiment. We note that
such a good agreement was observed in all
other experiments carried out on other heli-
copters. This creates confidence in the
reliability of the results obtained from
calculation and in the validity of the
initial assumptions, including that of the
permissibility of determining aerodynamic
forces by formulas based on the "steady-
state hypothesis" .
"flu rpm
■//////
/
/
/
uoa
/
/
/
/
1
1
i
1
- V
\
/
/
300
/I
200
>W/Vj
■////J
"m
/////,
ax
s
hift
of 2
II V
>>^
t
=0-
100
bal ancing
due to a
- n.86-kg weig
j
'""■--
^
f^
u,cj
23%
21%
25%
26%
'«//
Fig. 4. 39 Comparison of Experi-
mental and Calculated Data in
Flight.
It should be added that the flutter calculation pertains to a case quite
rare in rotor calculations when good agreement vd-th experiment is observed.
Probably, this is due primarily to the fact that even substantial errors in de-
termining the magnitudes of aerodynamic forces have no great effect on the final
results of calculation at critical flutter rpm.
479
4. Comparison of Calcula tion and Experiment in F light
Comparisons of calculation and experiment in flight do not show such good
agreement as in similar comparisons of results obtained under conditions of
axial flow past the rotor in ground tests. In flight, the decrease in critical
flutter rpm is felt more strongly than on the basis of calculation. Figure 4«39
gives two curves obtained by calculation for a regime with axial flow (ij, = O)
and for horizontal flight with p. = 0.25. The curves do not differ greatly.
Conversely, the experimental results differ substantially. In Fig. 4. 39 point 1
marks the critical rpm obtained in a ground test with a 0.86-kg weight attached
to the flaps while point 2 refers to the critical flutter rpm obtained on the
same helicopter in flight but without weights on the flaps. The test points in
Fig. 4. 39 were obtained in tests laid out by S.B.Bren and A.A.Dokuchayev and per-
formed by the pilot V.V.Vinitskiy.
The diagram indicates that the difference between the flight and ground
tests is appreciably greater than that obtained by calculation. The cause for
the difference lies in the fact that, in calculations, the amplitude of the /443
forced motion in the feathering hinge was taken to be the same on the ground and
in flight, i.e., it was assumed that in ground tests the amplitude of the angular
velocities of blade vibration in the feathering hinge, as a result of deflecting
the control stick, was the same as in flight as a consequence of ordinary flap-
ping motion. Here, it was disregarded that, in flight, the different vibrations
and oscillations with harmonics of higher orders may noticeably reduce the ef-
fectiveness of damping of oscillations due to friction in the feathering hinge.
This assumption is usually made to explain the more abrupt drop in critical
flutter rpm in flight in comparison with the calculation.
5. Check for Flutter
It has been noted above that, for a reliable elimination of the possibility
of flutter under service conditions, the helicopter rotor should have a well-
defined flutter margin. This margin should be checked on the ground and, if the
margin is below some standard value, the helicopter should not be allowed to fly.
In this approach, the required margin before flutter, checked on the ground,
should take into account a decrease in critical rpm in flight, possible deterio-
ration in flutter characteristics due to moisture penetrating into the blade,
and other factors, and should secure the necessary stability of blade vibration
at maximum approach to this margin.
The idea of flutter checking was first expressed by M.L.Mil' who proposed
to excite rotor oscillations by installing an eccentric in the cyclic pitch con-
trol system and to measxire the stability margin in terms of the amplitude of the
obtained resonance vibrations, which should be greater the smaller the flutter
margin. Such experiments were carried out and yielded interesting results.
Figure 4.40 shows the experimentally obtained dependence of the amplitude
of the hinge moment on the excitation frequency of the eccentric for various
rotor rpm. The diagram shows that the' higher the rotor rpm and hence the closer
to flutter, the greater will be the amplitude of the hinge moment. The same de-
pendence is obtained for blade balancing. Dm^ing experiments on the Mi-4 heli-
480
HO
n^l78rpm
copter vri-th a four-blade rotor, we noted the occurrence of two modes of resonance
vibrations of frequencies Pi = Pgco + n and Ps = Pecc - n (pgoc is the frequency
of excitation from the eccentric^, which agrees nicely with the theoretical
notions presented in Section 4» The experiments confirm the possibility of using
the described method for checking the stability margin of the rotor.
However, some time later a simpler method for checking the necessary margin
in terms of blade balancing was developed. This method provides for checking
the helicopter on the ground with blades whose balancing is shifted rearward by
a certain predetermined quantity.
The balance is shifted by means
of special weights placed on the
trailing edges of the blade
during the check. If, on raising
the rpm to a prescribed maximum,
flutter does not set in, the
weights are removed and the heli-
copter is admitted to service.
The weights were originally
selected on the basis of /hhh
calculations and later corrected
for different experiments and
service conditions. Two magni-
tudes of the required margins are
usually established. When the
helicopter is released from the
plant, an increased margin is
established -vdiich can be partial-
ly expended in service. There-
fore, in a number of cases flutter
check is also introduced in
service, but then a smaller re-
300 too p^^osc/min
Fig. 4. 40 Hinge Moment Amplitude as a
Function of Vibration Frequency of the
Eccentric.
quired margin is established.
The introduction of a flutter check has proved a useful measure, after
which cases of the development of flutter in service completely stopped.
6. Experimental Determination of Control System Rigidity
It was already pointed out above that the critical flutter rpm greatly de-
pends on the magnitude of the control system rigidity. It can be approximately
assumed that the critical flutter rpm is directly proportional to jCaon • Hence
it is obvious that it is important to determine control rigidity as accurately
as possible for a successful calculation. How does one determine the magnitude
of this rigidity? When performing the first calculations for flutter, control
rigidity is often calculated theoretically by stmming the design rigidities of
all components entering the control loop. First measurements of this rigidity
showed that the calculated values are much higher than the experimental values.
Therefore, it was necessary to reject the calculation of control rigidity.
481
However, the problem of the manner of experimental determination of control
rigidity also proved difficult. At first, the control rigidity was determined
statically, i.e., by the slope of the dependence of the magnitude of deformations
on the external load. However, this method did not clarify the mode of account-
ing for play in the control system, friction, and inertia of the components
entering this system. Therefore, the so-called dynamic method of determining
control rigidity was used, in which the external forces exerted at the control
by the blades were applied dynamically, at a frequency equal or close to the
frequency of flutter. With this method of measurement, the control rigidity /kh5
was by a factor of 2 - 2.5 less than with the static method.
It is natural that the results obtained in static analysis cannot be used
for the flutter calculations.
What is the simplest way of determining the dynamic rigidity of the control
system? For this, we used the following method:
On a helicopter with a nonrotating rotor we replaced the blades by special
weights whose moments of inertia relative to the feathering hinge were equal to
the moments of inertia of the removed blades. By measuring the natural vibra-
tion frequency of this system, the magnitude of the corresponding hinge control
rigidity can be completely defined. These rigidities can obviously be calculated
by means of the formula
where
Ccon-P'h^p, (8.2)
p = one of the natiiral vibration frequencies of this system, which
should be considered equivalent to a rotor with blades absolutely
rigid in torsion;
Coon ~ hinge control rigidity corresponding to the vibration mode for
which the frequency p is determined.
The necessary values of the natural vibration frequencies can be determined
by the usual method of forced vibrations with excitation by a vibrator or ec-
centric .
Since the rigidity of the longitudinal and lateral controls on a helicopter
is usually not the same, two different values of the natural vibration frequency
will correspond to loading of these controls [see eq.(4.19)].
Let us present the values of the frequencies corresponding to loading of
different control loops obtained on the Mi -4 helicopter with a nonrotating rotor:
/»^=400^420 osc/mi»
/7j=440-^450 osc/min
p^^ =590-1- 620 «wc/«/rt
p =920-^940<?J?<;/w/n.
482
The notations used here are the same as those used in eqs.(4«19)»
This raises the question whether the control rigidity thus measured depends
on the amplitude of external forces acting on the control system. To check this,
we carried out experiments with the maximum pennissible (in terms of strength)
magnitudes of hinge moments acting on the control, approximately the same as
those which act at the maximum flying speed, and witH moments lower by a factor
of 10. There was no substantial difference in the value of the obtained fre-
quencies.
Dynamic control rigidity may depend on the frequency of the forces acting
in the control cables. By changing the moments of inertia of the weights in-
stalled in place of the blades and measuring the new natural vibration frequen-
cies of the system, it becomes possible to define the mode of variation of
rigidity with variation of the vibration frequency. Figure 4.41 shows the re-
sults of such measurements. The abscissa gives the natural vibration frequency
for the control system which varies as a
function of the magnitude of the moment of
Px'P^x-pe^f osc/min^ ^ ^ ^ ^ inertia of the weights, while the ordinate
gives the dynamic rigidity expressed in
terms of the corresponding natural vibra-
tion frequency in agreement with eqs.(4.19).
Figure 4.41 indicates the approximate
values of the frequencies of variable /446
forces acting' in the nonrotating parts of
the lateral and longitudinal controls
during flutter with modes of the first
(m = l) and third order (m = 3). These
results illustrate the above assumption
(see Sect .4.3) that the magnitude of con-
trol rigidity may depend on the frequency
200 100 too posc/m/n of the forces acting in it.
200
Fig. 4. 41 Control Rigidity as a
Function of Vibration Frequency.
of effective forces.
For comparison. Fig. 4. 41 also gives
the values of the static control rigidity
obtained from the slope of the dependence
of control deformations on the magnitude
The dynamic method described here for determining control rigidity has been
sufficiently checked and can be recommended for practical use.
7. Experiments on Dynamically Similar Models
For conducting experiments on full-scale helicopters, the researcher usual-
ly runs into many difficulties having to do with observance of safety rules,
since full-scale experiments are usually carried out by a pilot or mechanic in
the helicopter. This imposes certain restrictions, especially for flutter tests
in flight where, for safety considerations, flutter is usually generated only
once in some regime or, in the extreme case, three to four times but never more
4^3
often. It is impossible to obtain any dependences for the parameter.
Furthermore, there are limitations to the possibility of investigating
various flight regimes, due to the characteristics of the helicopter on which
the experiment is carried out. The engineer is almost always interested in the
flutter margin with respect to rpm. However, the maximum rpa achievable in ex-
periments is limited by the capabilities of the engine. For example, the maxi-
mum flying speed is limited. Therefore, the researcher naturally will attempt
to make wind-tunnel tests on dynamically similar models. Such tests often jrield
interesting results. However, their wide use is restricted by a number of basic
shortcomings. To estimate the need for such tests in each individual case, let
us discuss the basic principles underlying the simulation in greater detail.
In producing a reduced-scale rotor model, geometric similitude of the ex-
ternal blade shape and the characteristic linear dimensions of rotor blade and
hub are of prime importance. We are thinking here of linear dimensions deter-
mining the planform of the blade; distribution of profiles and their setting
angles over the blade length; dimensions of its components determining, for
example, the position of the feathering hinge axis along the blade length; /447
relative position of other hub hinges; and many other dimensions. Next, it is
necessary that all relations between aerodynamic, inertia, and elastic forces
remain constant. In this case, the variable aerodynamic loads set up at the
model blade lead to the same relative deformations as on the original blade.
Let us examine this in greater detail for the example of bending vibrations
of a blade in the flapping plane. It can be demonstrated that bending deforma-
tions of a blade with respect to some natioral vibration harmonic are determined
by the coefficients of deformation calculated by the formula (see Vol. II)
Pj
where
pj = frequency of the j-th harmonic of natural blade bending vibration;
Yj = mass characteristic of the blade in vibrations of the j-th har-
monic [see eq.(7.55) of Chapt.I in Vol.11]
-^'^;e*o.7/?'
Y;=-^— ; (8.4)
mj
ck'j = dimensionless coefficient, characterizing the magnitude of work
done by the aerodynamic forces in displacements of the blade during
deformation with respect to the j-th harmonic:
a^ = JPt/<^'afr. (8.5)
Let us define the mode of variation in the relative coefficients of blade
bending deformations 6^^^ upon a similar change in all its geometric dimensions.
484
J
The relation between aerodynamic and inertia parameters of the blade is de-
termined by the values of the mass characteristics of the blade Yj • If all geo-
metric dimensions of the blade change the same number of times, namely, Kl times,
then, as follows from eq.(8.4), the mass characteristics of the blade do not
change.
However, we see from eq.(8.3) that, to retain similitude in bending de-
formations, the relation between the natural vibration frequency pj and the
angular velocity of rotation of the rotor cu must be retained. This requirement
is equivalent to keeping the Strouhal number constant:
where
p = vibration frequency;
U = velocity of flow.
The natural vibration frequency pj is determined by the formula
n2- 1
P-=-
lEl{yJdr-^^N(yJdr
(8.7)
Upon a similar change in all geometric dimensions of the blade, the quantity
of the elastic moment of inertia of its section I changes K* times. In this
case, as easily seen from eq.(8.7)> the magnitude of the natural vibration fre-
quency of the nonrotating blade po, changes Kl times. Consequently, the rela-
tion between this frequency and the angular velocity of rotation remains con- /kiS
stant if the angular velocity changes the same number of times.
Thus, to retain similitude in aerodynamic, inertia, and elastic forces, all
geometric blade dimensions must change the same number of times (Kl) and the
peripheral blade speeds must remain constant. Such dynamically similar models
are called Mach-similar models since similarity with respect to the Mach number
is retained in all blade sections.
The requirement of changing all geometric dimensions the same number of
times is easiest to meet by keeping the blade design unchanged. Therefore, the
development of such models actually reduces to the development of models similar
in design. This is a difficult problem, requiring the solution of many highly
complex technical problems and the organization of a special production of small-
dimension designs. A sufficiently high accuracy is necessary in their manu-
facture. Considerable difficulties also arise in developing hub hinges. It is
necessary to state that such models are also under considerable stress relative
to mechanical strength and do not permit much widening of the regimes in which
investigations can be carried out, in comparison with those on full-scale heli-
copters.
Upon a reduction of the geometric blade dimensions, the relation between
the blade weight and its aerodynamic and elastic characteristics drops by Ki.
times. This leads to a reduction of the influence of the blade weight para-
meters in comparison with the value for a full-scale helicopter. In particular,
IS5
the relative overhang of the blade of a nonrotating rotor decreases hy Ki times.
The blade, so to speak, becomes more rigid "to the eye". However, this dis-
turbance of similitude is observed only when the rotor is not rotating. Upon
rotation of the rotor the effect of the weight forces is generally negligible.
Therefore, a disturbance of their similitude has practically no effect on the
behavior of the blades.
The difficulties in developing Mach-similar and design-similar blades re-
sulted in their being used infrequently. Most often, dynamically similar blades
are developed with disturbance in similitude relative to the Mach ntm±ier. The
peripheral blade speeds on a model are reduced in comparison with the full-
scale blade by several times. In so doing, to retain the ratio of natural blade
vibration frequency pj to angular velocity of rotation cu, the blade rigidities
are reduced not by Kt times, as is required by geometric similitude, but by a
greater nimiber of times, most often by Kl. In this case, the necessary ratio of
natural vibration pj to angular velocity is achieved at peripheral speeds ^K^
smaller than those on a full-scale helicopter. Presumably, the results of tests
on such models can be extrapolated in totality to full-scale units only at
M < 0.4 (see Fig. 4*3) • At M = 0.5 - 0,9, the test results of such models can be
used only for qualitative estimates. In this connection, non-Mach-similar models
are used in only a limited volume for practical purposes.
486
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Translated for the National Aeronautics and Space Administration by the
O.W.Leibiger Research Laboratories, Inc.
F-49'+ NASA-Langley, 1967 2 489
ft
"Tke aeronautical and space activities of the United States shall be
conducted so as to contribute . . . to the expansion of human knowl-
edge of phenomena in the atmosphere and space. The Administration
shall provide for the widest practicable and appropriate dissemination
of information concerning its activities and the results thereof."
— Nauonal Abronauxics and Space Act of 1958
NASA SCIENTIFIC AND TECHNICAL PUBLICATIONS
TECHNICAL REPORTS: Scientific and technical information considered
important, complete, and a lasting contribution to existing knowledge.
TECHNICAL NOTES: Information less broad in scope but nevertheless of
importance as a contribution to existing knowledge.
TECHNICAL MEMORANDUMS: Information receiving limited distribu-
tion because of preliminary data, security classification, or other reasons.
CONTRACTOR REPORTS: Scientific and technical information generated
under a NASA contract or grant and considered an important contribution to
existing knowledge.
TECHNICAL TRANSLATIONS: Information published in a foreign
language considered to merit NASA distribution in English.
SPECIAL PUBLICATIONS: Information derived from or of value to NASA
activities. Publications include conference proceedings, monographs, data
compilations, handbooks, sourcebooks, and special bibliographies.
TECHNOLOGY UTILIZATION PUBLICATIONS: Information on tech-
nology used by NASA that may be of particular interest in commercial and other
non-aerospace applications. Publications include Tech Briefs, Technology
Utilization Reports and Notes, and Technology Surveys.
Details on the availability of these publications may be obtained from:
SCIENTIFIC AND TECHNICAL INFORMATION DIVISION
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Washington, D.C. 20546
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